Chapter 6

Simplify. $$ \frac{x^{4}+4 x^{3}-4 x^{2}+5 x}{x-5} $$

Use synthetic substitution to find \(f(-3)\) and \(f(4)\) for each function. \(f(x)=x^{3}-5 x^{2}+16 x-7\)

The graph of the polynomial function \(f(x)=a x(x-4)(x+1)\) goes through thepoint at \((5,15) .\) For what value(s) of \(x\) will \(f(x)=0 ?\)

OPEN ENDED. Write a polynomial of degree 5 that has three terms.

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(-2) $$

Graph each function. $$ y=\frac{1}{3}(x+5)^{2}-1 $$

CITY PLANNING. City planners have laid out streets on a coordinate grid before beginning construction. One street lies on the line with equation \(y=2 x+1 .\) Another street that intersects the first street passes through the point \((2,-3)\) and is perpendicular to the first street. What is the equation of the line on which the second street lies?

Simplify. $$ \left(3 x^{2}-2 x y+y^{2}\right)+\left(x^{2}+5 x y-4 y^{2}\right) $$

Use synthetic substitution to find \(f(-3)\) and \(f(4)\) for each function. \(f(x)=x^{4}+11 x^{3}-3 x^{2}+2 x-5\)

The graph of the polynomial function \(f(x)=a x(x-4)(x+1)\) goes through thepoint at \((5,15) .\) Simplify and rewrite the function as a cubic function.

CHECK FACTORING. Use a graphing calculator to determine if each polynomial is factored correctly. Write yes or no. If the polynomial is not factored correctly, find the correct factorization. $$ 3 x^{2}-48 \stackrel{?}{=} 3(x+4)(x-4) $$

Which one Doesn't Belong? Identify the expression that does not belong with the other three. Explain your reasoning. $$ 3 x y+6 x^{2} \quad \frac{5}{x^{2}} \quad x+5 \quad 5 b+11 c-9 a d^{2} $$

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(2) $$

Graph each function. $$ y=\frac{1}{2} x^{2}+x+\frac{3}{2} $$

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ x^{2}+7 x+8=0 $$

Simplify. $$ (2 x+4)(7 x-1) $$

Factor completely. If the polynomial is not factorable, write prime. \(15 a^{2} b^{2}-5 a b^{2} c^{2}\)

OPEN ENDED. Give an example of an equation that is not quadratic but can be written in quadratic form. Then write it in quadratic form.

CHALLENGE. What is the degree of the product of a polynomial of degree 8 and a polynomial of degree 6 ? Include an example to support your answer.

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(2 a) $$

Evaluate each determinant. $$ \left|\begin{array}{rr}{3} & {0} \\ {2} & {-2}\end{array}\right| $$

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ 3 x^{2}-9 x+2=0 $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \left[\begin{array}{rr}{3} & {6} \\ {2} & {-1}\end{array}\right] \cdot\left[\begin{array}{l}{a} \\\ {b}\end{array}\right]=\left[\begin{array}{c}{-3} \\ {18}\end{array}\right] $$

Factor completely. If the polynomial is not factorable, write prime. \(12 p^{2}-64 p+45\)

CHALLENGE. Factor \(64 p^{2 n}+16 p^{n}+1\)

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(a+1) $$

Evaluate each determinant. $$ \left|\begin{array}{rrr}{1} & {0} & {-3} \\ {2} & {-1} & {4} \\ {-3} & {0} & {2}\end{array}\right| $$

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ 2 x^{2}+3 x+2=0 $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \left[\begin{array}{rr}{5} & {-7} \\ {-3} & {4}\end{array}\right] \cdot\left[\begin{array}{c}{m} \\\ {n}\end{array}\right]=\left[\begin{array}{r}{-1} \\ {1}\end{array}\right] $$

Factor completely. If the polynomial is not factorable, write prime. \(4 y^{3}+24 y^{2}+36 y\)

REASONING. Find a counterexample to the statement \(a^{2}+b^{2}=(a+b)^{2}\)

ACT/SAT Which polynomial has degree 3\(?\) A \(x^{3}+x^{2}-2 x^{4}\) B \(-2 x^{2}-3 x+4\) C \(x^{2}+x+12^{3}\) D \(1+x+x^{3}\)

Solve each system of equations. $$ \begin{array}{l}{x+y=5} \\ {x+y+z=4} \\ {2 x-y+2 z=-1}\end{array} $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \begin{array}{l}{3 j+2 k=8} \\ {j-7 k=18}\end{array} $$

In a recent season, Monique Currie of the Duke Blue Devils scored 635 points. She made a total of 356 shots, including 3-point field goals, 2-point field goals, and 1-point free throws. She made 76 more 2-point field goals than free throws and 77 more free throws than 3-point field goals. Find the number of each type of shot she made.

Which polynomial represents \(\left(4 x^{2}+5 x-3\right)(2 x-7) ?\) F. \(8 x^{3}-18 x^{2}-41 x-21\) G. \(8 x^{3}+18 x^{2}+29 x-21\) H. \(8 x^{3}-18 x^{2}-41 x+21\) J. \(8 x^{3}+18 x^{2}-29 x+21\)

CHALLENGE. Explain how you would solve \(|a-3|^{2}-9|a-3|=-8 .\) Then solve the equation.

REVIEW \(\left(-4 x^{2}+2 x+3\right)-3\left(2 x^{2}-5 x+1\right)=\) \(\mathbf{F} \quad 2 x^{2}\) \(\mathbf{G}-10 x^{2}\) \(\mathbf{H}-10 x^{2}+17 x\) \(\mathbf{J} \quad 2 x^{2}+17 x\)

Solve each system of equations. $$ \begin{array}{l}{a+b+c=6} \\ {2 a-b+3 c=16} \\ {a+3 b-2 c=-6}\end{array} $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \begin{array}{l}{5 y+2 z=11} \\ {10 y-4 z=-2}\end{array} $$

Find all values of \(\pm \frac{a}{b}\) given each replacement set. \(a=\\{1,5\\} ; b=\\{1,2\\}\)

Simplify. \(\left(t^{3}-3 t+2\right) \div(t+2)\)

Simplify. Assume that no variable equals \(0 .\) $$ \left(-4 d^{2}\right)^{3} $$

Find all values of \(\pm \frac{a}{b}\) given each replacement set. \(a=\\{1,2\\} ; b=\\{1,2,7,14\\}\)

Simplify. \(\left(y^{2}+4 y+3\right)(y+1)^{-1}\)

ACT/SAT Which is not a factor of \(x^{3}-x^{2}-2 x ?\) A \(x\) B \(x+1\) C \(x-1\) D \(x-2\)

Simplify. Assume that no variable equals \(0 .\) $$ 5 r t^{2}(2 r t)^{2} $$

Find the greatest common factor of each set of numbers. $$ 18,27 $$

Find all values of \(\pm \frac{a}{b}\) given each replacement set. \(a=\\{1,3\\} ; b=\\{1,3,9\\}\)

Simplify. \(\frac{x^{3}-3 x^{2}+2 x-6}{x-3}\)