5.5 Zeros of Polynomial Functions

Solve: $9 m^{3} + 100 m = 60 m^{2} .$

Solve quadratic and higher order equations by factoring.

$x^{3} - x^{2} - 4 x + 4 = 0$

Use the Remainder Theorem to evaluate $f \left(x\right) = 2 x^{5} - 3 x^{4} - 9 x^{3} + 8 x^{2} + 2$ at $x = - 3.$

Use the Factor Theorem to find the zeros of $f \left(x\right) = x^{3} + 4 x^{2} - 4 x - 16$ given that $\left(x - 2\right)$ is a factor of the polynomial.

List all possible rational zeros of $f \left(x\right) = 2 x^{4} - 5 x^{3} + x^{2} - 4.$

Use the Rational Zero Theorem to find the rational zeros of $f \left(x\right) = 2 x^{3} + x^{2} - 4 x + 1.$

Use the Rational Zero Theorem to find the rational zeros of $f \left(x\right) = x^{3} - 5 x^{2} + 2 x + 1.$

Find the zeros of $f \left(x\right) = 2 x^{3} + 5 x^{2} - 11 x + 4.$

Find a fourth degree polynomial with real coefficients that has zeros of –3, 2, $i ,$ such that $f \left(−2\right) = 100.$

Find a third degree polynomial with real coefficients that has zeros of 5 and $- 2 i$ such that $f \left(1\right) = 10.$

Use Descartes’ Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for $f \left(x\right) = 2 x^{4} - 10 x^{3} + 11 x^{2} - 15 x + 12.$ Use a graph to verify the numbers of positive and negative real zeros for the function.

A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?

Describe a use for the Remainder Theorem.

What is the difference between rational and real zeros?

If synthetic division reveals a zero, why should we try that value again as a possible solution?

Use the Remainder Theorem to find the remainder.

$\left(3 x^{3} - 2 x^{2} + x - 4\right) \div \left(x + 3\right)$

Use the Remainder Theorem to find the remainder.

$\left(- 3 x^{2} + 6 x + 24\right) \div \left(x - 4\right)$

Use the Remainder Theorem to find the remainder.

$\left(x^{4} - 1\right) \div \left(x - 4\right)$

Use the Remainder Theorem to find the remainder.

$\left(4 x^{3} + 5 x^{2} - 2 x + 7\right) \div \left(x + 2\right)$

Use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$f \left(x\right) = 2 x^{3} + x^{2} - 5 x + 2 ; x + 2$

Use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$f \left(x\right) = 2 x^{3} + 3 x^{2} + x + 6 ; x + 2$

Use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$x^{3} + 3 x^{2} + 4 x + 12 ; x + 3$

Use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$2 x^{3} + 5 x^{2} - 12 x - 30 , ​ 2 x + 5$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} + 7 x^{2} - 10 x - 24 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$x^{3} + 5 x^{2} - 16 x - 80 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} - 3 x^{2} - 32 x - 15 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} - 3 x^{2} - x + 1 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} - 5 x^{2} + 9 x - 9 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 12 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$4 x^{4} + 4 x^{3} - 25 x^{2} - x + 6 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$x^{4} + 2 x^{3} - 4 x^{2} - 10 x - 5 = 0$

Use the Rational Zero Theorem to find the real solution(s) to each equation.

$8 x^{^{4}} + 26 x^{3} + 39 x^{2} + 26 x + 6$

Find all complex solutions (real and non-real).

$x^{3} - 8 x^{2} + 25 x - 26 = 0$

Find all complex solutions (real and non-real).

$3 x^{3} - 4 x^{2} + 11 x + 10 = 0$

Find all complex solutions (real and non-real).

$2 x^{3} - 3 x^{2} + 32 x + 17 = 0$

List all possible rational zeros for the functions.

$f \left(x\right) = 2 x^{^{3}} + 3 x^{2} - 8 x + 5$

List all possible rational zeros for the functions.

$f \left(x\right) = 6 x^{4} - 10 x^{2} + 13 x + 1$

Use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f \left(x\right) = 6 x^{3} - 7 x^{2} + 1$

Use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f \left(x\right) = 8 x^{3} - 6 x^{2} - 23 x + 6$

Use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f \left(x\right) = 16 x^{4} - 24 x^{3} + x^{2} - 15 x + 25$

Construct a polynomial function of least degree possible using the given information.

Real roots: –1, 1 (with multiplicity 2 and 1) and $\left(2 , f \left(2\right)\right) = \left(2 , 4\right)$

Construct a polynomial function of least degree possible using the given information.

Real roots: $- \frac{1}{2}$ , 0, $\frac{1}{2}$ and $\left(- 2 , f \left(- 2\right)\right) = \left(- 2 , 6\right)$

Find the dimensions of the box described.

The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

Find the dimensions of the box described.

The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

Find the dimensions of the box described.

The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

Find the dimensions of the right circular cylinder described.

The height is one less than one half the radius. The volume is $72 \pi$ cubic meters.

Find the dimensions of the right circular cylinder described.

The radius and height differ by two meters. The height is greater and the volume is $28.125 \pi$ cubic meters.

Use the Remainder Theorem to evaluate $f \left(x\right) = 6 x^{4} - x^{3} - 15 x^{2} + 2 x - 7$ at $x = 2.$

Show that $\left(x + 2\right)$ is a factor of $x^{3} - 6 x^{2} - x + 30.$ Find the remaining factors. Use the factors to determine the zeros of the polynomial.

Find the zeros of $f \left(x\right) = 4 x^{3} - 3 x - 1.$

Find the zeros of $f \left(x\right) = 3 x^{3} + 9 x^{2} + x + 3.$