Lial - College Algebra 13th Edition - Chapter 4

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Ch. 3 - Polynomial and Rational Functions

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 57

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = x^2 - 2x + 2; k = 1-i

Problem 57

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4-x^3+3x^2-8x+8; no real zero greater than 2

Problem 57

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

Problem 57

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. See Example 4. Zero of -3 having multiplicity 3; ƒ(3)=36

Problem 58

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=2x^5-x^4+2x^3-2x^2+4x-4; no real zero greater than 1

Problem 58

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)^2

Problem 58

Identify any vertical, horizontal, or oblique asymptotes in the graph of $y=f\left(x\right)$ . State the domain of $f$ .

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Problem 59

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4+x^3-x^2+3; no real zero less than -2

Problem 59

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = x^2 + 3x + 4; k = 2+i

Problem 60

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

$ƒ (x) = x^{5} + 2 x^{3} - 2 x^{2} + 5 x + 5$ ; no real zero less than -1

Problem 61

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 5+i and 5-i

Problem 61

Graph each rational function. See Examples 5–9. ƒ(x)=(x+1)/(x-4)

Problem 61

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = 4x^4 + x^2 + 17x + 3; k= -3/2

Problem 61

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x^4+2x^3-4x^2+x-1; no real zero greater than 1

Problem 62

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

$ƒ (x) = 3 x^{4} + 2 x^{3} - 4 x^{2} + x - 1$ ; no real zero less than -2

Problem 63

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = x^3 + 3x^2 -x + 1; k = 1+i

Problem 63

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5-3x^3+x+2; no real zero greater than 2

Problem 63

Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-3)

Problem 64

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

$ƒ (x) = x^{5} - 3 x^{3} + x + 2$ ; no real zero less than -3

Problem 65

The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x^3 - 2x^2 - x+2. Use the remainder theorem to find each of the following. Then determine the coor-dinates of the corresponding point on the graph of ƒ(x). ƒ (-2)

Problem 65

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Problem 66

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5.

$\frac{5}{x + 3} \geq \frac{3}{x}$

Problem 67

Graph each rational function. See Examples 5–9. ƒ(x)=3x/(x^2-x-2)

Problem 67

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 2-i, 3, and -1

Problem 68

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5.

$\frac{5}{2 - x} > \frac{3}{3 - x}$

Problem 69

The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x^3 - 2x^2 - x+2. Use the remainder theorem to find each of the following. Then determine the coor-dinates of the corresponding point on the graph of ƒ(x). ƒ (1)

Problem 69

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Problem 70

Graph each rational function. See Examples 5–9.

$ƒ (x) = x / (4 - x^{2})$

Problem 71

Graph each rational function. See Examples 5–9.

$ƒ (x) = [(x + 6) (x - 2)] / [(x + 3) (x - 4)]$

Problem 72

Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by $s (t) = - 16 t^{2} + 64 t + 100$ . Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?