# Practice Problems - Polynomials and Rational Functions

1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished.

Use synthetic division to find the function value.

1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2).

Write the quadratic function in the form y = a(x - h)2 + k.

2) y = x2 - 2x - 9

Find the product.

3) [x - (-6 + âˆš13)][x - (-6 - âˆš13)]

Use the leading coefficient test to determine whether yâ†’âˆž or yâ†’-âˆž as xâ†’ -âˆž.

4) y = -5x3 + 4x2 + 6x - 7

For the given function, find all asymptotes of the type indicated (if there are any).

5) f(x) = (x-9)/(x^2- 4) vertical

Use the rational zero theorem to find all possible rational zeros for the polynomial function.

6) P(x) = 3x3 + 43x2 + 43x + 27

Solve the inequality. Give answer in interval notation.

7) (x + 2)(x - 1)(x - 10) > 0

Solve the inequality.

8) (x+21)/(x+3) <2

Discuss the symmetry of the graph of the polynomial function.

9) f(x) = x2 + 2x - 1

Solve the absolute value equation.

10) |x2 - 10| = 4

Solve the quadratic inequality by graphing an appropriate quadratic function.

11) x2 - 2x - 8 â‰¤ 0

Use the theorem on bounds to establish the best integral bounds for the roots of the equation.

12) 6x3 - 7x2 + 7x + 9 = 0

State the degree of the polynomial equation.

13) 4(x + 8)2(x - 8)3 = 0

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1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished.

Let us use synthetic division to determine if the given set of numbers are zeros of the given polynomial

{-3, 2} f(x) = 3x^3 + 5x^2 - 6x + 18.

ï€3 3 5 -6 18

-9 12 -18

3 ï€4 6 0

Since the last number (remainder) is 0, x = -3 is a zero of f(x) = 0 ïƒž x + 3 is a factor of f(x)

2 3 5 -6 18

6 22 32

3 11 16 50

Since the last number (remainder) is nonzero, x = 2 is not a zero of f(x) = 0 ïƒž x - 2 is a not factor of f(x) ANSWERS:

Synthetic division has been illustrated.

Use synthetic division to find the function value.

1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2).

-2 2 4 2 3 8

ïƒŸ -4 0 -4 2

2 0 2 -1 10

ANSWERS:

f(-2) = Remainder = 10

Write the quadratic function in the form y = a(x - h)2 + k.

2) y = x2 - 2x - 9

y = x^2 - 2x - 9

y = x^2 - 2x + 4 - 13

y = (x - 2)^2 - 13

This is of the form y = a(x - h)^2 + k, where

a = 1, h = 2, k = -13 ANSWERS:

y = (x - 2)^2 - 13

Find the product.

3) [x - (-6 + âˆš13)][x - (-6 - âˆš13)]

Given expression = (x + 6 - Ã–13)(x + 6 + Ã–13)

= (x + 6)^2 - (Ã–13)^2

= x^2 + 12x + 36 - 13

= x^2 + 12x + 23 ANSWERS:

Use the leading coefficient test to determine whether yâ†’âˆž or yâ†’-âˆž as xâ†’ -âˆž.

4) y = -5x3 + 4x2 + 6x - 7

Since the leading coefficient is negative, as x ïƒ¨ -âˆž,

y ïƒ¨ âˆž ANSWERS:

y ïƒ¨ âˆž

For the given function, find all asymptotes of the type indicated (if there are any).

5) f(x) = (x-9)/(x^2- 4) vertical

f(x) = (x - 9)/(x + 2)(x - 2)

f(x) has vertical tangents at x = {-2, 2} ANSWERS:

x = -2 and x = 2

Use the rational zero theorem to find all possible rational zeros for the polynomial function.

6) P(x) = 3x3 + 43x2 + 43x + 27

The possible rational zeros of P(x) are given by

Â± (factors of 27)/factors of 3

ïƒž Â± (1, 3, 9, 27)/(1, 3)

ïƒž {Â± 1, Â± 3, Â± 9, Â± 27, Â± 1/3} ANSWERS:

{Â± 1, Â± 3, Â± 9, Â± 27, Â± 1/3}

Solve the inequality. Give answer in interval notation.

7) (x + 2)(x - 1)(x - 10) > 0

Given expression = (x + 2)(x - 1)(x - 10)

The critical numbers are x = {-2, 1, 10}

For x < -2, (x + 2)(x - 1)(x - 10) < 0

For -2 < x < 1, (x + 2)(x - 1)(x - 10) > 0

For 1 < x < 10, (x + 2)(x - 1)(x - 10) < 0

For x > 10, (x + 2)(x - 1)(x - 10) > 0

ïƒž The solution is (-2, 1) Ãˆ (10, âˆž) ANSWERS:

(-2, 1) Ãˆ (10, âˆž)

Solve the inequality.

8) (x+21)/(x+3) <2

(x + 21/(x + 3) < 2

(x + 21)(x + 3) < 2(x + 3)^2

0 < (x + 3)[2(x + 3) - (x + 21)]

0 < (x + 3)(x - 15)

(x + 3)(x - 15) > 0

ïƒž x < -3 or x > 15 ANSWERS:

(-âˆž, -3) Ãˆ (15, âˆž)

Discuss the symmetry of the graph of the polynomial function.

9) f(x) = x2 + 2x - 1

f(-x) = x^2 - 2x - 1 and -f(x) = -x^2 - 2x + 1

= -(x^2 + 2x - 1)

We see that f(-x) â‰ f(x) and also f(-x) â‰ -f(x) ANSWERS:

f(x) has no symmetry of any kind.

Solve the absolute value equation.

10) |x2 - 10| = 4

|x^2 - 10| = 4 ïƒž x^2 - 10 = Â± 4

x^2 = 10 Â± 4

x^2 = 14 or 6

x = Ã–14 or x = Ã–6 ANSWERS:

x = {Â± Ã–6, Â± Ã–14}

Solve the quadratic inequality by graphing an appropriate quadratic function.

11) x2 - 2x - 8 â‰¤ 0

ANSWERS:

The solution is [-2, 4]

Use the theorem on bounds to establish the best integral bounds for the roots of the equation.

12) 6x3 - 7x2 + 7x + 9 = 0

The possible rational factors of

f(x) = 6x^3 - 7x^2 + 7x + 9 are given by

{Â± (factors of 9)/factors of 6}

= {Â± (1, 3, 9)/(1, 2, 3, 6)}

= {Â± 1, Â± 3, Â± 9, Â± Â½, Â± 3/2, Â± 9/2, Â± 1/3, Â± 1/6}

There are 2 sign changes in f(x) ïƒž f(x) has 0 positive roots or 2 positive roots. In fact f(x) has two complex roots.

f(-x) = -6x^3 - 7x^2 - 7x + 9

There is just one sign change in f(-x) ïƒž f(x) has one negative root

From the list of possible roots, choose x = 1/6 and

x = -1

-1 6 -7 7 9

ïƒŸ -6 13 -20

6 -13 20 -11

Since the entries in the last row alternate in sign,

x = -1 is a lower bound for the roots of f(x)

-1/2 6 -7 7 9

ïƒŸ -3 5 -6

6 -10 12 3

x = -1/2 is an upper bound ANSWERS:

x = -1 is a lower bound and x = -1/2 is an upper bound.

State the degree of the polynomial equation.

13) 4(x + 8)2(x - 8)3 = 0

4(x + 8)^2 (x - 8)^3 = 0 has a degree 2 + 3 = 5 ANSWERS:

The degree is 5.

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