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Precalculus Study Guide

Chapter 3: Graphing Polynomial Functions

Introduction

After straight lines, the simplest functions to study are the real polynomials, the general form of which is

$f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + ... + x_{2} x^{2} + a_{1} x + a_{0}$

where the real numbers $a_{k}, k = 0, 1, ..., n$ , are the coefficients, with $a_{n}$ called the leading coefficient. Real polynomials are those whose coefficients are real numbers. The integer $n$ is the degree of the polynomial.

If $n = 2$ , the polynomial function, which then has the form

$f (x) = a_{2} x^{2} + a_{1} x + a_{0}$

is called quadratic, whereas if $n = 3$ , the polynomial function, which then has the form

$f (x) = a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$

is called cubic.

If $f (r) = 0$ , then $x = r$ is called a zero of the polynomial, and $x - r$ is then a factor of the polynomial. For each zero $r_{k}$ , there is a corresponding factor $x - r_{k}$ , so that the factored form of the polynomial is

$f (x) = (x - r_{1}) (x - r_{2}) \dots (x - r_{n})$

Hence, given the zeros of a polynomial, it is possible to reconstruct the polynomial itself.

(The fundamental theorem of algebra states that every polynomial of degree $n$ has exactly $n$ zeros, although they may not all be distinct - there can be repeated zeros.)

If the polynomial function is graphed, the real zeros are the $x$ -intercepts.

Chapter Glossary

The following terms in Chapter 3 are linked to the Maple Math Dictionary.

coefficient

complete the square

complex conjugate

coordinate

cubic

factor

fundamental theorem of algebra

integer

intercept

leading coefficient

long division

monic polynomial

parabola

polynomial

quadratic formula

quartic

real number

synthetic division

tangent
zero

Typical Problems

For the polynomial functions given in Problems 1 - 3, sketch a graph and determine both the $y$ - and $x$ -intercepts. Since the $x$ -intercepts are the zeros of the polynomials, they can be found approximately from the graph, or exactly, by factoring the polynomial over the real numbers.

3.1. (a) $f (x) = 3 x^{2} - 5 x - 2$

(b) $f (x) = 2 x^{2} - 4 x + 2$

3.2. (a) $f (x) = {(x + 1)}^{3}$

(b) $f (x) = x^{3} - 5 x^{2} + 4 x + 10$

(d) $f (x) = x^{3} - 2 x^{2} - x + 2$

3.3. (a) $f (x) = x^{4} - 12 x^{3} + 47 x^{2} - 62 x + 26$

(b) $f (x) = x^{4} - 15 x^{3} + 75 x^{2} - 135 x + 74$

3.4. For each of the following, find the real monic polynomial of least degree having the given numbers as its zeros. (A real polynomial is one whose coefficients are real. A monic polynomial has leading coefficient 1, that is, the coefficient of the term with the highest power is 1.)

(a) $\{- 2, 3\}$

(b) { $- 2$ + $i$ , 1 }

(d) $\{3 - 4 i, 7\}$

Maple Initializations

Before accessing the Maple version of the solutions to the typical problems stated above, initialize Maple by pressing the button provided on the right.

Solutions

Problem 3.1

3.1 - Mathematical Solution

3.1 (a) - Mathematical Solution

For the function $f (x) = 3 x^{2} - 5 x - 2$ , the $y$ -intercept is found to be $f (0) = - 2$ , and the $x$ -intercepts are found from the quadratic formula to be

$x_{1} = \frac{- (- 5) + \sqrt{{(- 5)}^{2} - 4 (3) (- 2)}}{2 (3)}$ = 2

and

$x_{2} = \frac{- (- 5) - \sqrt{{(- 5)}^{2} - 4 (3) (- 2)}}{2 (3)}$ = $- \frac{1}{3}$

Moreover, $f (x)$ factors to the product $(3 x + 1) (x - 2)$ .

Since the function $f (x)$ is a quadratic polynomial, its graph is that of a parabola, opening upward. The standard form of this parabola would be

Figure 3.1.1 Graph of $f (x) = 3 x^{2} - 5 x - 2$

$y + \frac{49}{12} = 3 {(x - \frac{5}{6})}^{2}$

obtained by completing the square in $x$ . Thus, the coordinates of the vertex of the parabola are $(\frac{5}{6}, - \frac{49}{12})$ .

From this information, Figure 3.1.1, a graph of the function, can be drawn.

3.1 (b) - Mathematical Solution

For the function $f (x) = 2 x^{2} - 4 x + 2$ , the $y$ -intercept is found to be $f (0) = 2$ , and the $x$ -intercepts are found from the quadratic formula to be

$x_{1} = \frac{- (- 4) + \sqrt{{(- 4)}^{2} - 4 (2) (2)}}{2 (2)}$ = 1

and

$x_{2} = \frac{- (- 4) - \sqrt{{(- 4)}^{2} - 4 (2) (2)}}{2 (2)}$ = $1$

Moreover, $f (x)$ factors to the product $2 {(x - 1)}^{2}$ .

Since the function $f (x)$ is a quadratic polynomial, its graph is that of a parabola, opening upward. The standard form of this parabola would be

Figure 3.1.2 Graph of $f (x) = 2 x^{2} - 4 x + 2$

$y = 2 {(x - 1)}^{2}$

obtained by completing the square in $x$ . Thus, the coordinates of the vertex of the parabola are $(1, 0)$ .

From this information, Figure 3.1.2, a graph of the function, can be drawn.

3.1 (c) - Mathematical Solution

For the function $f (x) = x^{2} - 2 x + 5$ , the $y$ -intercept is found to be $f (0) = 5$ , and the $x$ -intercepts are found from the quadratic formula to be

$x_{1} = \frac{- (- 2) + \sqrt{{(- 2)}^{2} - 4 (1) (5)}}{2 (1)}$ = $1 + 2 i$

and

$x_{2} = \frac{- (- 2) - \sqrt{{(- 2)}^{2} - 4 (1) (5)}}{2 (1)}$ = $1 - 2 i$

Moreover, $f (x)$ does not factor over the real numbers.

Since the function $f (x)$ is a quadratic polynomial, its graph is

Figure 3.1.3 Graph of $f (x) = x^{2} - 2 x + 5$

that of a parabola, opening upward. The standard form of this parabola would be

$y - 4 = {(x - 1)}^{2}$

obtained by completing the square in $x$ . Thus, the coordinates of the vertex of the parabola are $(1, 4)$ .

From this information, Figure 3.1.3, a graph of the function, can be drawn.

3.1 - Maplet Solution

3.1 (a) - Maplet Solution

A graph of the polynomial function $f (x) = 3 x^{2} - 5 x - 2$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.1.4.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.1.4 Thumbnail image of Polynomial Tutor #1

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.1 (b) - Maplet Solution

A graph of the polynomial function $f (x) = 2 x^{2} - 4 x + 2$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.1.5.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.1.5 Thumbnail image of Polynomial Tutor #1

Since the $x$ -intercepts are the zeros of the polynomial, if a zero is repeated, it will appear the appropriate number of times in the list of $x$ -intercepts.

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.1 (c) - Maplet Solution

A graph of the polynomial function $f (x) = x^{2} - 2 x + 5$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.1.6.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.1.6 Thumbnail image of Polynomial Tutor #1

If the polynomial has no real zeros, its graph will not cross the $x$ -axis. Hence, there will be no $x$ -intercepts returned.

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.1 - Interactive Solution

3.1 (a) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fa$

Obtain Figure 3.1.1

•	Type $fa$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 1 \leq x \leq 3$

Obtain $x$ -intercept(s)

•	Type $fa = 0$ and press the Enter key.

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fa$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.1 (b) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fb$

Obtain Figure 3.1.2

•	Type $fb$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 1 \leq x \leq 3$

Obtain $x$ -intercept(s)

•	Type $fb = 0$ and press the Enter key.

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fb$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.1 (c) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fc$

Obtain Figure 3.1.3

•	Type $fc$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 1 \leq x \leq 3$

Obtain $x$ -intercept(s)

•	Type $fc = 0$ and press the Enter key.

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fc$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.1 - Programmatic Solution

3.1 (a) - Programmatic Solution

Enter the polynomial.

>	$Ya ≔ 3 x^{2} - 5 x - 2$

Graph the polynomial as per Figure 3.1.1.

>	$plot (Ya, x = - 1 .. 3)$

Obtain the $x$ -intercepts.

>	$solve (Ya = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Ya, x = 0)$

3.1 (b) - Programmatic Solution

Enter the polynomial.

>	$Yb ≔ 2 x^{2} - 4 x + 2$

Graph the polynomial as per Figure 3.1.1.

>	$plot (Yb, x = - 1 .. 3)$

Obtain the $x$ -intercepts.

>	$solve (Yb = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Yb, x = 0)$

3.1 (c) - Programmatic Solution

Enter the polynomial.

>	$Yc ≔ x^{2} - 2 x + 5$

Graph the polynomial as per Figure 3.1.1.

>	$plot (Yc, x = - 1 .. 3)$

Obtain the $x$ -intercepts.

>	$solve (Yc = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Yc, x = 0)$

Problem 3.2

3.2 - Mathematical Solution

3.2 (a) - Mathematical Solution

The polynomial $f (x) = {(x + 1)}^{3}$ has $f (0) = 1$ for its $y$ -intercept, and $x = - 1$ for its $x$ -intercept.

The graph is a translation to the left, by one unit, of the graph of $x^{3}$ . Hence, the graph of $f (x)$ can be sketched, as shown in Figure 3.2.1.

Figure 3.2.1 Graph of $f (x) = {(x + 1)}^{3}$

3.2 (b) - Mathematical Solution

The polynomial $f (x) = x^{3} - 5 x^{2} + 4 x + 10$ has $y$ -intercept $f (0) = 10$ , and $x$ -intercept $x = - 1$ . This is found by searching for integer zeros, computing the values of $f (k), k = - 3, - 2, ...$ .

Although there are formulas expressing the zeros of a cubic in terms of its coefficients, they are very cumbersome to apply. That $x = - 1$ is the only zero of $f (x)$ is determined by dividing out the factor $x + 1$ , leaving the quadratic polynomial $x^{2} - 6 x + 10$ whose zeros are the complex conjugate pair 3 + $i$ .

The division of $f (x)$ by the factor $x + 1$ can be done by synthetic division, the tableau for which appears in Table 3.2.1

Figure 3.2.2 Graph of $f (x) = x^{3} - 5 x^{2} + 4 x + 10$

The final zero is the remainder, so the factor divides evenly. The coefficients of the depressed polynomial are the numbers to the left of the final zero in the bottom row of the tableau.

The zeros of the depressed polynomial are found with the quadratic formula, and they are

1 $- 5$ 4 10 [-1

$- 1$ 6 $- 10$

________________

1 $- 6$ 10 0

Table 3.2.1 Synthetic division

$x_{1} = \frac{- (- 6) + \sqrt{{(- 6)}^{2} - 4 (1) (10)}}{2}$ = $\frac{6 + \sqrt{- 4}}{2} = \frac{6 + 2 i}{2}$ = $3 + i$

$x_{2} = \frac{- (- 6) - \sqrt{{(- 6)}^{2} - 4 (1) (10)}}{2}$ = $\frac{6 - \sqrt{- 4}}{2} = \frac{6 - 2 i}{2}$ = $3 - i$

Without a plotting device such as a calculator or computer, a graph can only be drawn from computed values such as those appearing in Table 3.2.2.

$x$	$- 3.0$	$- 2.5$	$- 2.0$	$- 1.5$	$- 1.0$	$- &period; 5$	$0.$
$f (x)$	$- 74.0$	$- 46.9$	$- 26.0$	$- 10.6$	$0.$	$6.62$	$10.0$

$x$	$0.5$	$1.0$	$1.5$	$2.0$	$2.5$	3.0
$f (x)$	$10.9$	$10.0$	$8.12$	$6.0$	$4.38$	$4.0$
Table 3.2.2 Values of the function $f (x) = x^{3} - 5 x^{2} + 4 x + 10$

The graph of $f (x)$ in Figure 3.2.2 was obtained with Maple. Drawing such a graph with just a pencil and paper is a daunting task.

3.2 (c) - Mathematical Solution

The polynomial $f (x) = x^{3} - 3 x + 2$ has $y$ -intercept $f (0) = 2$ and an $x$ -intercept at $x = - 2$ . This is found by searching for zeros amongst the integers, and fortunately, this zero is an integer. Thus, $x - (- 2) = x + 2$ is a factor of the polynomial.

The companion factor is found by dividing $f (x)$ by $x + 2$ . This division can be done by long division or by the synthetic division shown in the tableau in Table 3.2.3.

The final zero in the last row of the tableau is the remainder, so the factor divides evenly. The coefficients of the depressed polynomial are the numbers to the left of the final zero in the bottom row of the tableau.

Figure 3.2.3 Graph of $f (x) = x^{3} - 3 x + 2$

The zeros of the depressed polynomial are found with the quadratic formula or by noticing that $x^{2} - 2 x + 1 = {(x - 1)}^{2}$ . Thus, the remaining zeros are 1 and again 1, so that $x = 1$ is a double zero. There are two $x$ -intercepts, namely, $x = - 2$ and $x = 1$ , the latter being a double zero of the polynomial.

1 0 $- 3$ 2 [ $- 2$

$- 2$ $4$ $- 2$

_________________

1 $- 2$ 1 0

Table 3.2.3 Synthetic division

From this information, a sketch of $f (x)$ can be drawn. The graph in Figure 3.2.3 was obtained in Maple.

3.2 (d) - Mathematical Solution

The polynomial $f (x) = x^{3} - 2 x^{2} - x + 2$ has $y$ -intercept $f (0) = 2$ and an $x$ -intercept at $x = - 1$ . This is found by searching for zeros amongst the integers, and fortunately, this zero is an integer. Thus, $x - (- 1) = x + 1$ is a factor of the polynomial.

The companion factor is found by dividing $f (x)$ by $x + 1$ . This division can be done by long division or by the synthetic division shown in the tableau in Table 3.2.4.

Figure 3.2.4 Graph of $f (x) = x^{3} - 2 x^{2} - x + 2$

The zeros of the depressed polynomial $x^{2} - 3 x + 2$ are found with the quadratic formula, and they are

1 $- 2$ $- 1$ 2 [ $- 1$

$- 1$ $4$ $- 2$

_________________

1 $- 3$ 2 0

Table 3.2.4 Synthetic division

$x_{1} = \frac{- (- 3) + \sqrt{{(- 3)}^{2} - 4 (1) (2)}}{2}$ = $\frac{3 + \sqrt{1}}{2} = \frac{3 + 1}{2}$ = $2$

$x_{2} = \frac{- (- 3) - \sqrt{{(- 3)}^{2} - 4 (1) (2)}}{2}$ = $\frac{3 - \sqrt{1}}{2} = \frac{3 - 1}{2}$ = $1$

Thus, the remaining $x$ -intercepts are $x = 1$ and $x = 2$ , so a sketch of $f (x)$ could now be drawn. The graph in Figure 3.2.4 was obtained in Maple.

3.2 - Maplet Solution

3.2 (a) - Maplet Solution

A graph of the polynomial function $f (x) = {(x + 1)}^{3}$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.2.5.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.2.5 Thumbnail image of Polynomial Tutor #1

Since the $x$ -intercepts are the zeros of the polynomial, if a zero is repeated, it will appear the appropriate number of times in the list of $x$ -intercepts.

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.2 (b) - Maplet Solution

A graph of the polynomial function $f (x) = x^{3} - 5 x^{2} + 4 x + 10$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail in Figure 3.2.6.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.2.6 Thumbnail image of Polynomial Tutor #1

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.2 (c) - Maplet Solution

A graph of the polynomial function $f (x) = x^{3} - 3 x + 2$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.2.7.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.2.7 Thumbnail image of Polynomial Tutor #1

Since the $x$ -intercepts are the zeros of the polynomial, if a zero is repeated, it will appear the appropriate number of times in the list of $x$ -intercepts.

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.2 (d) - Maplet Solution

A graph of the polynomial function $f (x) = x^{3} - 2 x^{2} - x + 2$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.2.8.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.2.8 Thumbnail image of Polynomial Tutor #1

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.2 - Interactive Solution

3.2 (a) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fa$

Obtain Figure 3.2.1

•	Type $fa$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 3 \leq x \leq 1$

Obtain $x$ -intercept(s)

•	Type $fa = 0$ and press the Enter key.

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fa$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.2 (b) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fb$

Obtain Figure 3.2.2

•	Type $fb$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 3 / 2 \leq x \leq 4$

Obtain $x$ -intercept(s)

•	Type $fb = 0$ and press the Enter key.

•	Context Panel: Solve≻Numerically Solve

Obtain the $y$ -intercept

Type $fb$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.2 (c) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fc$

Obtain Figure 3.2.3

•	Type $fc$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 3 \leq x \leq 3$

Obtain $x$ -intercept(s)

•	Type $fc = 0$ and press the Enter key.

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fc$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.2 (d) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fd$

Obtain Figure 3.2.4

•	Type $fd$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 2 \leq x \leq 3$

Obtain $x$ -intercept(s)

•	Type $fd = 0$ and press the Enter key.

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fd$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.2 - Programmatic Solution

3.2 (a) - Programmatic Solution

Enter the polynomial.

>	$Ya ≔ {(x + 1)}^{3}$

Graph the polynomial as per Figure 3.2.1.

>	$plot (Ya, x = - 3 .. 1, y = - 4 .. 4)$

Obtain the $x$ -intercepts.

>	$solve (Ya = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Ya, x = 0)$

3.2 (b) - Programmatic Solution

Enter the polynomial.

>	$Yb ≔ x^{3} - 5 x^{2} + 4 x + 10$

Graph the polynomial as per Figure 3.2.2

>	$plot (Yb, x = - 2 .. 5, y = - 10 .. 12)$

Obtain the $x$ -intercepts.

>	$remove (has, [solve (Yb = 0, x)], I) []$

Obtain the $y$ -intercept.

>	$eval (Yb, x = 0)$

3.2 (c) - Programmatic Solution

Enter the polynomial.

>	$Yc ≔ x^{3} - 3 x + 2$

Graph the polynomial as per Figure 3.2.3

>	$plot (Yc, x = - 3 .. 3, y = - 10 .. 10)$

Obtain the $x$ -intercepts.

>	$solve (Yc = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Yc, x = 0)$

3.2 (d) - Programmatic Solution

Enter the polynomial.

>	$Yd ≔ x^{3} - 2 x^{2} - x + 2$

Graph the polynomial as per Figure 3.2.4

>	$plot (Yd, x = - 2 .. 3)$

Obtain the $x$ -intercepts.

>	$solve (Yd = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Yd, x = 0)$

Problem 3.3

3.3 - Mathematical Solution

3.3 (a) - Mathematical Solution

The polynomial $f (x) = x^{4} - 12 x^{3} + 47 x^{2} - 62 x + 26$ has $f (0) = 26$ for its $y$ -intercept, and $x = 1$ for one of its $x$ -intercepts. This intercept is found by searching amongst the integers for zeros of the polynomial. Since $x = 1$ is a zero, the factor $x - 1$ divides the polynomial evenly. The companion factor is found by either long division or synthetic division. The tableau in Table 3.3.1 implements synthetic division by the factor $x - 1$ .

1 $- 12$ 47 $- 62$ 26 [ 1

1 $- 11$ $36$ $- 26$

___________________________

1 $- 11$ 36 $- 26$ 0

Table 3.3.1 Synthetic division

Figure 3.3.1 Graph of $f (x) = x^{4} - 12 x^{3} + 47 x^{2} - 62 x$

The companion factor, the "depressed polynomial," is $x^{3} - 11 x^{2} + 36 x - 26$ , which also has $x = 1$ as a zero. Again, this is found by searching through the integers. Once again, the factor $x - 1$ can be divided out, this time from the depressed polynomial. The synthetic division tableau for this division appears in Table 3.3.2.

1 $- 11$ 36 $- 26$ [ 1

1 $- 10$ 26

______________________

1 $- 10$ $- 26$ 0

Table 3.3.2 Synthetic division

The new depressed polynomial is $x^{2} - 10 x - 26$ , whose zeros can be found by the quadratic formula. These remaining zeros are

$x_{1} = \frac{- (- 10) + \sqrt{{(- 10)}^{2} - 4 (1) (26)}}{2}$ = $\frac{10 + \sqrt{- 4}}{2} = 5 + i$

$x_{2} = \frac{- (- 10) - \sqrt{{(- 10)}^{2} - 4 (1) (26)}}{2}$ = $\frac{10 - \sqrt{- 4}}{2} = 5 - i$

Knowledge of just the $y$ -intercept and the double zero at $x = 1$ does not yet lead to an unambiguous graph. Even making use of the property of quartics that their graphs will at most resemble the letter "W" still leaves a need for a list of values such as found in Table 3.3.3.

$x$	$- 1.0$	$- 0.5$	$0.0$	$0.5$	$1.0$	$1.5$	$2.0$	$2.5$	$3.0$	$3.5$	$4.0$	$4.5$	$5.0$	$5.5$	$6.0$
$f (x)$	$148.0$	$70.3$	$26.0$	$5.3$	$0.0$	3.3	$10.0$	$16.3$	$20.0$	$20.3$	$18.0$	$15.3$	$16.0$	$25.3$	$50.0$
Table 3.3.3 Values of the function $f (x) = x^{4} - 12 x^{3} + 47 x^{2} - 62 x$

On the basis of all this information, the sketch of $f (x)$ seen in Figure 3.3.1 could now be drawn. (However, Figure 3.3.1 was, in fact, generated by Maple.)

3.3 (b) - Mathematical Solution

The polynomial $f (x) = x^{4} - 15 x^{3} + 75 x^{2} - 135 x + 74$ has $f (0) = 74$ for its $y$ -intercept, and $x = 1$ for one of its $x$ -intercepts. This intercept is found by searching amongst the integers for zeros of the polynomial. Since $x = 1$ is a zero, the factor $x - 1$ divides the polynomial evenly. The companion factor is found by either long division or synthetic division. The tableau in Table 3.3.4 implements synthetic division by the factor $x - 1$ .

1 $- 15$ 75 $- 135$ 74 [ 1

1 $- 14$ $61$ $- 74$

___________________________

1 $- 14$ 61 $- 74$ 0

Table 3.3.4 Synthetic division

Figure 3.3.2 Graph of $f (x) = x^{4} - 15 x^{3} + 75 x^{2} - 135 x + 74$

The companion factor, the "depressed polynomial," is $x^{3} - 14 x^{2} + 61 x - 74$ , which has $x = 2$ as a zero. Again, this is found by searching through the integers. The factor $x - 2$ can be divided out from the depressed polynomial. The synthetic division tableau for this division appears in Table 3.3.5.

1 $- 14$ 61 $- 74$ [ 2

2 $- 24$ 74

______________________

1 $- 12$ $37$ 0

Table 3.3.5 Synthetic division

The new depressed polynomial is $x^{2} - 12 x + 37$ , whose zeros can be found by the quadratic formula. These remaining zeros are

$x_{1} = \frac{- (- 12) + \sqrt{{(- 12)}^{2} - 4 (1) (37)}}{2}$ = $\frac{12 + \sqrt{- 4}}{2} = 6 + i$

$x_{2} = \frac{- (- 12) - \sqrt{{(- 12)}^{2} - 4 (1) (37)}}{2}$ = $\frac{12 - \sqrt{- 4}}{2} = 6 - i$

Knowledge of just the $y$ -intercept and the zeros at $x = 1$ and $x = 2$ does not yet lead to an unambiguous graph. Even making use of the property of quartics that their graphs will at most resemble the letter "W" still leaves a need for a list of values such as the one given in Table 3.3.6.

$x$	$f (x)$	$x$	$f (x)$	$x$	$f (x)$	$x$	$f (x)$
$- 1.0$	$300.0$	$1.5$	$- 5.3$	$4.0$	$30.0$	$6.5$	$30.9$
$- 0.5$	$162.2$	$2.0$	$0.0$	$4.5$	$28.4$	$7.0$	$60.0$
$0.0$	$74.0$	$2.5$	$9.9$	$5.0$	$24.0$	$7.5$	$116.2$
$0.5$	$23.4$	$3.0$	$20.0$	$5.5$	$19.7$	$8.0$	210.0
$1.0$	$0.0$	$3.5$	$27.2$	$6.0$	$20.0$
Table 3.3.6 Values of $f (x) = x^{4} - 15 x^{3} + 75 x^{2} - 135 x + 74$

On the basis of all this information, the sketch of $f (x)$ seen in Figure 3.3.2 could now be drawn. (However, Figure 3.3.2 was, in fact, generated by Maple.)

3.3 - Maplet Solution

3.3 (a) - Maplet Solution

A graph of the polynomial function $f (x) = x^{4} - 12 x^{3} + 47 x^{2} - 62 x + 26$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.3.3.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.3.3 Thumbnail image of Polynomial Tutor #1

Since the $x$ -intercepts are the zeros of the polynomial, if a zero is repeated, it will appear the appropriate number of times in the list of $x$ -intercepts.

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.3 (b) - Maplet Solution

A graph of the polynomial function $f (x) = x^{4} - 15 x^{3} + 75 x^{2} - 135 x + 74$ , along with its intercepts, can be obtained with Polynomial Tutor #1 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.3.4.

To use this tutor, enter the coefficients of the polynomial in the appropriate boxes. Blanks will be interpreted as the number zero (0).

The Display Function button will display the polynomial entered, while the buttons labeled Graph, $x$ -intercepts, and $y$ -intercept will generate the corresponding results.

Figure 3.3.4 Thumbnail image of Polynomial Tutor #1

To launch Polynomial Tutor #1, click the following link: Polynomial Tutor #1

3.3 - Interactive Solution

3.3 (a) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fa$

Obtain Figure 3.3.1

•	Type $fa$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 1 \leq x \leq 6$ Options≻Range from: $0 \leq y \leq 30$

Obtain $x$ -intercept(s)

•	Type $fa$ and press the Enter key.

•	Context Panel: Factor

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fa$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.3 (b) - Interactive Solution

•	Enter the polynomial.

•	Context Panel: Assign to a Name≻ $fb$

Obtain Figure 3.3.1

•	Type $fb$ and press the Enter key.

•	Context Panel: Plots≻Plot Builder $- 1 \leq x \leq 6$ Options≻Range from: $- 5 \leq y \leq 140$

Obtain $x$ -intercept(s)

•	Type $fb$ and press the Enter key.

•	Context Panel: Factor

•	Context Panel: Solve≻Solve

Obtain the $y$ -intercept

Type $fb$ and press the Enter key.

Context Panel: Evaluate at a Point≻ $x = 0$

3.3 - Programmatic Solution

3.3 (a) - Programmatic Solution

Enter the polynomial.

>	$Ya ≔ x^{4} - 12 x^{3} + 47 x^{2} - 62 x + 26$

Graph the polynomial as per Figure 3.2.1.

>	$plot (Ya, x = - 1 .. 6, y = 0 .. 30)$

Factor the polynomial.

>	$factor (Ya)$

Obtain the $x$ -intercepts.

>	$solve (Ya = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Ya, x = 0)$

3.3 (b) - Programmatic Solution

Enter the polynomial.

>	$Yb ≔ x^{4} - 15 x^{3} + 75 x^{2} - 135 x + 74$

Graph the polynomial as per Figure 3.2.1.

>	$plot (Yb, x = - 1 .. 6, y = - 5 .. 140)$

Factor the polynomial.

>	$factor (Yb)$

Obtain the $x$ -intercepts.

>	$solve (Yb = 0, x)$

Obtain the $y$ -intercept.

>	$eval (Yb, x = 0)$

Problem 3.4

3.4 - Mathematical Solution

	3.4 (a) - Mathematical Solution
	If the zeros of a polynomial are $- 2$ and 3, then the polynomial's factored form is $(x - (- 2)) (x - 3)$ = $(x + 2) (x - 3)$ and its expanded form is $x^{2} - x - 6$

3.4 (b) - Mathematical Solution

If the zeros of a polynomial are $- 2$ + $i$ and 1, then the polynomial's factored form is

$(x - (- 2 + i)) (x - (- 2 - i)) (x - 1)$	= $(x + 2 - i) (x + 2 + i) (x - 1)$
	= $(x^{2} + 4 x + 5) (x - 1)$

and its expanded form is

$x^{3} + 3 x^{2} + x - 5$

3.4 (c) - Mathematical Solution

If the zeros of a polynomial are 1, 3, and $1$ + $2 i$ , then the polynomial's factored form is

$(x - (1 + 2 i)) (x - (1 - 2 i)) (x - 1) (x - 3)$	= $(x - 1 - 2 i) (x - 1 + 2 i) (x - 1) (x - 3)$
	= $(x^{2} - 2 x + 5) (x - 1) (x - 3)$

and its expanded form is

$x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 15$

3.4 (d) - Mathematical Solution

If the zeros of a polynomial with real coefficients are $3 - 4 i$ and $7$ , then there must be at least one other zero, namely, $3 + 4 i$ , the complex conjugate of the given zero $3 - 4 i$ . The factored form of the desired polynomial is then

$(x - (3 + 4 i)) (x - (3 - 4 i)) (x - 7)$	= $(x - 3 - 4 i) (x - 3 + 4 i) (x - 7)$
	= $(x^{2} - 6 x + 25) (x - 7)$

and its expanded form is

$x^{3} - 13 x^{2} + 67 x - 175$

3.4 - Maplet Solution

3.4 (a) - Maplet Solution

The real monic polynomial of least degree having $\{- 2, 3\}$ as its zeros can be found with Polynomial Tutor #2 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.4.1.

To use this tutor, enter the zeros in the appropriate box, and press the button labeled Linear factored form. This gives the polynomial as a product of its linear factors, that is, as $(x - (- 2)) (x - 3) = (x + 2) (x - 3)$ .

Pressing the button labeled Factored over reals displays the polynomial in terms of factors containing just real coefficients. In this problem where the zeros are all real, the linear factored form and this form are the same.

Figure 3.4.1 Thumbnail image of Polynomial Tutor #2

Pressing the button labeled Expanded form displays the polynomial as a sum of "coefficients times powers of $x$ ." Colloquially, this is the "multiplied out" form $x^{2} - x - 6$ .

As in Polynomial Tutor #1, there are buttons for producing a graph of the required polynomial, and buttons for displaying the intercepts.

To launch Polynomial Tutor #2, click the following link: Polynomial Tutor #2

3.4 (b) - Maplet Solution

The real monic polynomial of least degree having $\{- 2 \pm i, 1\}$ as its zeros can be found with Polynomial Tutor #2 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.4.2.

To use this tutor, enter the zeros in the appropriate box, using I for the complex unit $i = \sqrt{- 1}$ . Each complex zero must be entered separately, since there is no "+" operator in Maple.

Pressing the button labeled Linear factored form gives the polynomial as a product of its linear factors, that is, as

$(x - (- 2 + i)) (x - (- 2 - i)) (x - 3)$

$= (x + 2 - i) (x + 2 + i) (x - 3)$

Figure 3.4.2 Thumbnail image of Polynomial Tutor #2

Pressing the button labeled Factored over reals displays the polynomial as

$(x^{2} + 4 x + 5) (x - 1)$

a product of factors containing just real coefficients.

Pressing the button labeled Expanded form displays the polynomial as a sum of "coefficients times powers of $x$ ." Colloquially, this is the "multiplied out" form $x^{3} + 3 x^{2} + x - 5$ .

As in Polynomial Tutor #1, there are buttons for producing a graph of the required polynomial, and buttons for displaying the intercepts.

To launch Polynomial Tutor #2, click the following link: Polynomial Tutor #2

3.4 (c) - Maplet Solution

The real monic polynomial of least degree having $\{1, 3, 1 \pm 2 i\}$ as its zeros can be found with Polynomial Tutor #2 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.4.3.

To use this tutor, enter the zeros in the appropriate box, using I for the complex unit $i = \sqrt{- 1}$ . Each complex zero must be entered separately, since there is no "+" operator in Maple.

Pressing the button labeled Linear factored form gives the polynomial as a product of its linear factors, that is, as

$(x - 1) (x - 3) (x - (1 + 2 i)) (x - (1 - 2 i))$

$= (x - 1) (x - 3) (x - 1 - 2 i) (x - 1 + 2 i)$

Figure 3.4.3 Thumbnail image of Polynomial Tutor #2

Pressing the button labeled Factored over reals displays the polynomial as

$(x - 1) (x - 3) (x^{2} - 2 x + 5)$

a product of factors containing just real coefficients.

Pressing the button labeled Expanded form displays the polynomial as a sum of "coefficients times powers of $x$ ." Colloquially, this is the "multiplied out" form $x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 15$ .

As in Polynomial Tutor #1, there are buttons for producing a graph of the required polynomial, and buttons for displaying the intercepts.

To launch Polynomial Tutor #2, click the following link: Polynomial Tutor #2

3.4 (d) - Maplet Solution

The real monic polynomial of least degree having $\{3 - 4 i, 7\}$ as its zeros will also have $3 + 4 i$ as a zero. Remember, polynomials with real coefficients will have any of its complex zeros appearing as complex conjugate pairs. The required polynomial can be found with Polynomial Tutor #2 .

Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 3.4.4.

To use this tutor, enter the zeros in the appropriate box, using I for the complex unit $i = \sqrt{- 1}$ . Each complex zero must be entered separately, since there is no "+" operator in Maple.

Pressing the button labeled Linear factored form gives the polynomial as a product of its linear factors, that is, as

Figure 3.4.4 Thumbnail image of Polynomial Tutor #2

$(x - (3 + 4 i)) (x - (3 - 4 i)) (x - 7)$ $= (x - 3 - 4 i) (x - 3 + 4 i) (x - 7)$

Pressing the button labeled Factored over reals displays the polynomial as

$(x^{2} - 6 x + 25) (x - 7)$

a product of factors containing just real coefficients.

Pressing the button labeled Expanded form displays the polynomial as a sum of "coefficients times powers of $x$ ." Colloquially, this is the "multiplied out" form $x^{3} - 13 x^{2} + 67 x - 175$ .

As in Polynomial Tutor #1, there are buttons for producing a graph of the required polynomial, and buttons for displaying the intercepts.

To launch Polynomial Tutor #2, click the following link: Polynomial Tutor #2

3.4 - Interactive Solution

3.4 (a) - Interactive Solution

•	Enter the first zero.

•	Context Panel: Assign to a Name≻ $a$

•	Enter the second zero.

•	Context Panel: Assign to a Name≻ $b$

Write the polynomial as a product of linear factors.

Context Panel: Expand

3.4 (b) - Interactive Solution

•	Enter the first zero.

•	Context Panel: Assign to a Name≻ $a$

•	Enter the second zero.

•	Context Panel: Assign to a Name≻ $b$

•	Enter the third zero.

•	Context Panel: Assign to a Name≻ $c$

Write the polynomial as a product of linear factors.

Context Panel: Expand
Context Panel: Factor

3.4 (c) - Interactive Solution

•	Enter the first zero.

•	Context Panel: Assign to a Name≻ $a$

•	Enter the second zero.

•	Context Panel: Assign to a Name≻ $b$

•	Enter the third zero.

•	Context Panel: Assign to a Name≻ $c$

•	Enter the fourth zero.

•	Context Panel: Assign to a Name≻

Write the polynomial as a product of linear factors.

Context Panel: Expand
Context Panel: Factor

3.4 (d) - Interactive Solution

•	Enter the real zero.

•	Context Panel: Assign to a Name≻ $a$

•	Enter the complex zero.

•	Context Panel: Assign to a Name≻ $b$

•	Enter the third zero, the complex conjugate of $b$ .

•	Context Panel: Assign to a Name≻ $c$

Write the polynomial as a product of linear factors.

Context Panel: Expand
Context Panel: Factor

3.4 - Programmatic Solution

3.4 (a) - Programmatic Solution

Enter the given zeros, assigning them to the names $z_{k}, k = 1, 2$ .

>	$z_{1} ≔ - 2 &semi; z_{2} ≔ 3$

Write the desired polynomial as a product of linear factors.

>	$f ≔ (x - z_{1}) (x - z_{2})$

Expand the product of factors.

>	$expand (f)$

3.4 (b) - Programmatic Solution

Enter the given zeros, assigning them to the names $z_{k}, k = 1, 2$ .

>	$z_{1} ≔ - 2 + I &semi; z_{2} ≔ conjugate (z_{1}) &semi; z_{3} ≔ 1$

Write the desired polynomial as a product of linear factors.

>	$f ≔ (x - z_{1}) (x - z_{2}) (x - z_{3})$

Expand the product of factors.

>	$F ≔ expand (f)$

Factor the polynomial over the reals.

>	$factor (F)$

3.4 (c) - Programmatic Solution

Enter the given zeros, assigning them to the names $z_{k}, k = 1, 2$ .

>	$z_{1} ≔ 1 + 2 I &semi; z_{2} ≔ \overset{&conjugate0;}{z_{1}} &semi; z_{3} ≔ 1 &semi; z_{4} ≔ 3$

Write the desired polynomial as a product of linear factors.

>	$f ≔ \prod_{k = 1}^{4} (x - z_{k})$

Expand the product of factors.

>	$F ≔ expand (f)$

Factor the polynomial over the reals.

>	$factor (F)$

3.4 (d) - Programmatic Solution

Enter the given zeros, assigning them to the names $z_{k}, k = 1, 2$ .

>	$z_{1} ≔ 7 &semi; z_{2} ≔ 3 - 4 I &semi; z_{3} ≔ \overset{&conjugate0;}{z_{2}}$

Write the desired polynomial as a product of linear factors.

>	$f ≔ (x - z_{1}) (x - z_{2}) (x - z_{3})$

Expand the product of factors.

>	$F ≔ expand (f)$

Factor the polynomial over the reals.

>	$factor (F)$

Exercises - Chapter 3

For the quadratic polynomial functions given in Exercises 3.1 - 3.6, sketch a graph and determine both the $y$ - and $x$ -intercepts. Since the $x$ -intercepts are the zeros of the polynomials, they can be found by the quadratic formula.

3.1. $f (x) = x^{2} - 8 x + 12$

3.2. $f (x) = x^{2} + 2 x - 63$

3.3. $f (x) = x^{2} - 3 x - 10$

3.4. $f (x) = x^{2} - 2 x - 8$

3.5. $f (x) = 2 x^{2} - x + 5$

3.6. $f (x) = 7 x^{2} + 6 x - 8$

For the cubic polynomial functions given in Exercises 3.7 - 3.12, sketch a graph and determine both the $y$ - and $x$ -intercepts. Since the $x$ -intercepts are the zeros of the polynomials, they can be found approximately from the graph, or exactly, by factoring the polynomial over the real numbers.

3.7. $f (x) = x^{3} + 11 x^{2} + 20 x - 32$

3.8. $f (x) = x^{3} - 23 x^{2} + 175 x - 441$

3.9. $f (x) = x^{3} - 14 x^{2} + 39 x + 54$

3.10. $f (x) = x^{3} + 4 x^{2} - 29 x + 24$

3.11. $f (x) = 4 x^{3} - 8 x^{2} + 4 x - 8$

3.12. $f (x) = 6 x^{3} + x^{2} - 4 x + 1$

For the quartic polynomial functions given in Exercises 3.13 - 3.18, sketch a graph and determine both the $y$ - and $x$ -intercepts. Since the $x$ -intercepts are the zeros of the polynomials, they can be found approximately from the graph, or exactly, by factoring the polynomial over the real numbers.

3.13. $f (x) = x^{4} - 6 x^{3} - 45 x^{2} + 122 x + 360$

3.14. $f (x) = x^{4} - 6 x^{3} - 45 x^{2} + 274 x - 336$

3.15. $f (x) = x^{4} - 7 x^{3} - 24 x^{2} + 112 x + 128$

3.16. $f (x) = x^{4} - 13 x^{2} + 52 x - 40$

3.17. $f (x) = x^{4} - 13 x^{2} + 52 x - 40$

3.18. $f (x) = x^{4} - 9 x^{3} + 21 x^{2} + 29 x - 114$

In Exercises 3.19 - 3.20, one complex zero of a quartic polynomial whose coefficients are real is given. Find the remaining three zeros. Hint: If using some form of technology, simply find all zeros as in Exercises 3.13 - 3.18. Otherwise, find the quadratic factor corresponding to the given complex zero, and long divide to find the other quadratic factor, which will then yield to the quadratic formula. (Complex arithmetic is treated at greater length in Chapter 8.)

3.19. $f (x) = 6 x^{4} - 56 x^{3} + 406 x^{2} + 276 x + 148$ with zero $5 + 7 i$

3.20. $f (x) = x^{4} - 5 x^{3} + 29 x^{2} - 13 x - 116$ with zero $2 - 5 i$

In Exercises 3.21 - 3.24, find the real monic polynomial of least degree having the given numbers as its zeros.

3.21. $\{- 3, 2\}$

3.22. $\{- 2, 3, 3\}$

3.23. $\{2, 2, 3 - 5 i\}$

3.24. $\{1 + i, 2 - 3 i\}$

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