Chapter 10

The length \(\ell\) of a box is 3 inches less than the height \(h .\) The width \(w\) is 9 inches less than the height. The box has a volume of 324 cubic inches. What are the dimensions of the box?

$$ \text { Find the product }(2 x+1)(x+3) \text { using the } $$distributive property and explain how this leads to the FOIL pattern.

Write the number in decimal form. \(8.17 \times 10^{7}\)

Use linear combinations to solve the linear system. Then check your solution. $$ \begin{aligned} &2 x+y=120\\\ &x+2 y=120 \end{aligned} $$

Simplify the expression. $$ -4(1-x)+7 $$

In Exercises 62 and \(63,\) use the following information about a basketball player's hang time, the length of time spent in the air after jumping. The maximum height \(h\) jumped (in feet) is a function of \(t,\) where \(t\) is the hang time (in seconds). Hang time model: \(h=4 t^{2}\). If you jump 1 foot into the air, what is your hang time?

Solve the equation. $$ (x+12)(x+7)=0 $$

Which of the following is the complete factorization of \(x^{3}-5 x^{2}+4 x-20 ?\) A) \((x+2)(x+2)(x-5)\) B) \((x+2)(x-2)(x-5)\) C) \(\left(x^{2}+4\right)(x-5)\) D) \((x-4)(x-1)(x-20)\)

Simplify the expression. Use only positive exponents. $$x^{7} \cdot \frac{1}{x^{4}}$$

Find the product \(2 a^{2}\left(a^{2}-3 a+1\right)\) $$ \text { (A) } 2 a^{2}-6 a+2 $$ $$ \text { (B) } 2 a^{4}-6 a^{3}+2 a $$ $$ \text { (C) } 2 a^{2}-3 a^{3}+2 a^{2} $$ $$ \text { (D) } 2 a^{4}-6 a^{3}+2 a^{2} $$

Find the product. \((x-2)(x-7)\)

Find the product. $$ (4 t-1)^{2} $$

Simplify the expression. $$ -12 x-5(11-x) $$

In Exercises 62 and \(63,\) use the following information about a basketball player's hang time, the length of time spent in the air after jumping. The maximum height \(h\) jumped (in feet) is a function of \(t,\) where \(t\) is the hang time (in seconds). Hang time model: \(h=4 t^{2}\). If a professional player jumps 4 feet into the air, what is the hang time?

Solve the equation. $$ (z+2)(z+3)=0 $$

Simplify the expression. Use only positive exponents. $$\frac{5 x^{4} y}{3 x y^{2}} \cdot \frac{9 x y}{x^{2} y}$$

Solve \(x^{3}-4 x=0\) F) 0 and 2 G) \(0,2,\) and \(-2\) H) 2 and \(-2\) J) \(-2\) and 0

Find the product \((x+9)(x-2)\) $$(F) x^{2}+7 x-18$$ $$ \text { (G) } x^{2}-11 x-18 $$ $$ (\mathrm{H}) x^{2}-18 $$ $$ (\mathrm{J}) x^{2}-7 x $$

Find the product. \((x+8)(x-8)\)

Find the product. $$ (b+9)(b-9) $$

The table below shows mileage and gasoline used for 6 months. For each of these months, find the mileage rate in miles per gallon. Round to the nearest tenth. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \text { Mileage (miles) } & {295} & {320} & {340} & {280} & {310} & {355} \\ \hline \text { Gas Used (gallons) } & {12.3} & {13.3} & {14.2} & {11.6} & {12.9} & {14.8} \\ \hline \end{array} $$

In the sport of pole-vaulting, the height \(h\) (in feet) reached by a pole- vaulter can be approximated by a function of \(v,\) the velocity of the pole- vaulter, as shown in the model below. The constant \(g\) is approximately 32 feet per second per second. Pole-vaulter height model: \(h=\frac{v^{2}}{2 g}\) . To reach a height of 9 feet, what is the pole-vaulter's velocity?

Solve the equation. $$ (t-19)^{2}=0 $$

Sketch the graph of the function. Label the vertex. $$y=2 x^{2}+3 x+6$$

Find the product \((x-1)\left(2 x^{2}+x+1\right)\) $$ \text { (A) } 2 x^{3}-3 x^{2}-1 $$ $$ \text { (B) } 2 x^{3}-x^{2}-2 x-1 $$ $$ \text { (C) } 2 x^{3}-x^{2}-1 $$ $$ \text { (D) } 2 x^{3}+3 x^{2}+2 x+1 $$

Find the product. \((x-4)(x+5)\)

Find the product. $$ (3 x+5)(3 x+5) $$

In Exercises \(65-70,\) simplify. Then use a calculator to evaluate the expression. $$ 2^{2} \cdot 2^{3} $$

In the sport of pole-vaulting, the height \(h\) (in feet) reached by a pole- vaulter can be approximated by a function of \(v,\) the velocity of the pole- vaulter, as shown in the model below. The constant \(g\) is approximately 32 feet per second per second. Pole-vaulter height model: \(h=\frac{v^{2}}{2 g}\) . What height will a pole-vaulter reach if the pole-vaulter's velocity is 32 feet per second?

Solve the equation. $$ 5(x-9)(x-6)=0 $$

Sketch the graph of the function. Label the vertex. $$y=3 x^{2}-9 x-12$$

Simplify the expression. Write your answer as a power. $$ (7 x)^{2} $$

Find the product. \((2 x+7)(3 x-1)\)

Find the product. $$ (2 a-7)(2 a+7) $$

Which of the following is a correct factorization \( \text { of }-12 x^{2}+147 ?\) A. \(-3(2 x+7)^{2}\) B. \(3(2 x-7)(2 x+7)\) C. \(-2(2 x-7)(2 x+7)\) D. \(-3(2 x-7)(2 x+7)\)

In Exercises \(65-70,\) simplify. Then use a calculator to evaluate the expression. $$ \left(3^{2} \cdot 1^{3}\right)^{2} $$

Solve the equation. $$ (y+47)(y-27)=0 $$

Sketch the graph of the function. Label the vertex. $$y=-x^{2}+4 x+16$$

$$ \left(\frac{1}{3} m\right)^{2} $$

Find the product. \((5 x-1)(5 x+2)\)

Find the product. $$ (11-6 x)^{2} $$

Which of the following is a correct factorization of \(72 x^{2}-24 x+2 ?\) F. \(-9(3 x-1)^{2}\) G. \(2(6 x-1)^{2}\) H. \(8(3 x-1)^{2}\) J. \(9(3 x-1)^{2}\)

In Exercises \(65-70,\) simplify. Then use a calculator to evaluate the expression. $$ \left[(-1)^{8} \cdot 2^{4}\right]^{2} $$

Solve the equation. $$ (z-1)(4 z+2)=0 $$

Multiply the fractions. $$\frac{1}{2} \cdot \frac{1}{2}$$

Solve the equation. \(|x|=3\)

$$ \left(\frac{2}{5} y\right)^{2} $$

Find the product. \((3 x+1)(8 x-3)\)

Find the product. $$ (100+27 x)^{2} $$

Solve \(9 x^{2}-12 x+4=0\) A. \(-3\) B. \(-\frac{2}{3}\) C. \(\frac{2}{3}\) D. 3