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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 + √13)][x - (-6 - √13)]

Use the leading coefficient test to determine whether y→∞ or y→-∞ as x→ -∞.
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 ≤ 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.

3 3 5 -6 18
-9 12 -18
3 4 6 0
Since the last number (remainder) is 0, x = -3 is a zero of f(x) = 0  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  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
 -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 = ...

Solution Summary

All questions neatly solved showing the steps in detail. Graphs included.

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