Mid-Chapter Mixed Review

Determine whether the statement is true or false.

The y-intercept of the graph of the function $P (x) = 5 - 2 x^{3}$ $P (x) = 5 - 2 x^{3}$ is (5, 0). [4.2]
The degree of the polynomial $x - \frac{1}{2} x^{4} - 3 x^{6} + x^{5}$ $x - \frac{1}{2} x^{4} - 3 x^{6} + x^{5}$ is 6. [4.1]
If $f (x) = (x + 7) (x - 8)$ $f (x) = (x + 7) (x - 8)$ , then $f (8) = 0$ $f (8) = 0$ . [4.3]
If $f (12) = 0$ $f (12) = 0$ , then $x + 12$ $x + 12$ is a factor of $f (x)$ $f (x)$ . [4.3]

Find the zeros of the polynomial function and state the multiplicity of each. [4.1]

In Exercises 9–12, match the function with one of the graphs (a)–(d) that follow. [4.2]

Using the intermediate value theorem, determine, if possible, whether the function has at least one real zero between a and b. [4.2]

$f (x) = x^{3} - 2 x^{2} + 3; a = - 2, b = 0$ $f (x) = x^{3} - 2 x^{2} + 3; a = - 2, b = 0$
$f (x) = x^{3} - 2 x^{2} + 3; a = - \frac{1}{2}, b = 1$ $f (x) = x^{3} - 2 x^{2} + 3; a = - \frac{1}{2}, b = 1$
For the polynomial $P (x) = x^{4} - 6 x^{3} + x - 2$ $P (x) = x^{4} - 6 x^{3} + x - 2$ and the divisor $d (x) = x - 1$ $d (x) = x - 1$ , use long division to find the quotient $Q (x)$ $Q (x)$ and the remainder $R (x)$ $R (x)$ when $P (x)$ $P (x)$ is divided by $d (x)$ $d (x)$ . Express $P (x)$ $P (x)$ in the form $d (x) \cdot Q (x) + R (x)$ $d (x) \cdot Q (x) + R (x)$ . [4.3]

Use synthetic division to find the quotient and the remainder. [4.3]

$(3 x^{4} - x^{3} + 2 x^{2} - 6 x + 6) \div (x - 2)$ $(3 x^{4} - x^{3} + 2 x^{2} - 6 x + 6) \div (x - 2)$
$(x^{5} - 5) \div (x + 1)$ $(x^{5} - 5) \div (x + 1)$

Use synthetic division to find the function values. [4.3]

For $g (x) = x^{3} - 9 x^{2} + 4 x - 10$ $g (x) = x^{3} - 9 x^{2} + 4 x - 10$ , find $g (- 5)$ $g (- 5)$ .
For $f (x) = 20 x^{2} - 40 x$ $f (x) = 20 x^{2} - 40 x$ , find $f (\frac{1}{2})$ $f (\frac{1}{2})$ .
For $f (x) = 5 x^{4} + x^{3} - x$ $f (x) = 5 x^{4} + x^{3} - x$ , find $f (- \sqrt{2})$ $f (- \sqrt{2})$ .

Using synthetic division, determine whether the numbers are zeros of the polynomial function. [4.3]

$- 3 i, 3; f (x) = x^{3} - 4 x^{2} + 9 x - 36$ $- 3 i, 3; f (x) = x^{3} - 4 x^{2} + 9 x - 36$
$- 1, 5; f (x) = x^{6} - 35 x^{4} + 259 x^{2} - 225$ $- 1, 5; f (x) = x^{6} - 35 x^{4} + 259 x^{2} - 225$

Factor the polynomial function $f (x)$ $f (x)$ . Then solve the equation $f (x) = 0$ $f (x) = 0$ . [4.3]

$h (x) = x^{3} - 2 x^{2} - 55 x + 56$ $h (x) = x^{3} - 2 x^{2} - 55 x + 56$
$g (x) = x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 24$ $g (x) = x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 24$

Collaborative Discussion and Writing

How is the range of a polynomial function related to the degree of the polynomial? [4.1]
Is it possible for the graph of a polynomial function to have no y-intercept? no x-intercepts? Explain your answer. [4.2]
Explain why values of a function must be all positive or all negative between consecutive zeros. [4.2]
In synthetic division, why is the degree of the quotient 1 less than that of the dividend? [4.3]

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