Question

zeros of polynomial functions
sketch the graph of the function by finding the zeros. list the zeros.

1. $f(x) = 2x^3 - 12x^2 - 6x$ 2. $f(x) = x^3 - 2x^2 - 4x - 6$

(images of coordinate grids here)
find the zeros of each function and describe the behavior of the graph of the function at each zero.

1. $x^3 - 8x^2 + 18x$ 4. $x^3 + x^2 - 3x + 1$

determine all the real and complex zeros of each polynomial function.

1. $f(x) = x^3 - 7x^2 + 4x - 28$ 6. $f(x) = x^3 - x^2 - 2x + 8$
2. a company that sells toys models their profit with the function $p(x) = -4x^3 + 32x^2 - 64$. their profit $p$, in thousands of dollars, is a function of the number of toys sold $x$ measured in hundreds. what do the key features of the graph reveal about the profits? what is the maximum profit the company can make?

solve each inequality.

1. $x^3 - 27x < 0$ 9. $x^3 + 9x^2 - 10x > 0$
2. use your graphing calculator to determine if $f(x) = (x - 1)(x - 6)(x + 3)$ is the correct factorization of $f(x) = x^3 + 7x^2 + 4x - 12$. explain.

Explanation:

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Problem 1:

Step1: Factor out GCF

$f(x)=2x(x^2 - 6x - 3)$

Step2: Set to zero, solve for $x$

$2x=0 \implies x=0$; solve $x^2-6x-3=0$ using quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$:
$x=\frac{6\pm\sqrt{36+12}}{2}=\frac{6\pm\sqrt{48}}{2}=3\pm2\sqrt{3}$

Answer:

(Problem1):
Zeros: $x=0$, $x=3+2\sqrt{3}$, $x=3-2\sqrt{3}$

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Problem 2:

Step1: Use Rational Root Theorem

Test possible roots $\pm1,\pm2,\pm3,\pm6$. $x=3$ is a root: $3^3-2(3)^2-4(3)-6=27-18-12-6=-9
eq0$; $x=-1$: $-1-2+4-6=-5
eq0$; $x=3$ fails, use cubic formula or approximate:

Step2: Approximate real root

Using Newton-Raphson: $x\approx3.239$, complex roots: $x\approx-0.619\pm1.115i$