Chapter 6

Perform each indicated operation. Write all results in lowest terms. $$ \frac{3}{7} \cdot \frac{12}{17} $$

Factor. $$ x^{3}+125 $$

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 5 q^{2}-4 p q-5 q+4 p $$

Multiply. $$ (a+3 b)(9 a-4 b) $$

Factor each trinomial completely. See Examples 1 through 7. \(2 t^{4}+3 t^{2}-27\)

Explain the error and solve correctly: $$ \begin{array}{l} x(x-2)=8 \\ x=8 \text { or } x-2=8 \\ x=10 \end{array} $$

Factor. $$ x^{3}+216 $$

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 6 m^{2}-5 m n-6 m+5 n $$

Multiply. $$ (y-5 x)(6 y+5 x) $$

Factor each trinomial completely. See Examples 1 through 7. \(4 r^{4}-17 r^{2}-15\)

Explain the error and solve correctly: \((x-4)(x+2)=0\) \(x=-4\) or \(x=2\)

Factor. $$ 3 x^{6} y^{2}+81 y^{2} $$

Factor out the GCF from each polynomial. Then factor by grouping. $$ 12 x^{2} y-42 x^{2}-4 y+14 $$

Write a polynomial that factors as \((x-3)(x+8)\).

Write a quadratic equation that has two solutions, 6 and -1 . Leave the polynomial in the equation in factored form.

Factor. $$ x^{2} y^{9}+x^{2} y^{3} $$

Factor out the GCF from each polynomial. Then factor by grouping. $$ 90+15 y^{2}-18 x-3 x y^{2} $$

Write a quadratic equation that has two solutions, 0 and -2 . Leave the polynomial in the equation in factored form.

Solve each equation. $$ x-5=0 $$

Factor out the GCF from each polynomial. Then factor by grouping. $$ 6 a^{2}+9 a b^{2}+6 a b+9 b^{3} $$

Complete each sentence in your own words. If \(x^{2}+b x+c\) is factorable and \(c\) is negative, then the signs of the last-term factors of the binomials are opposite because...

Write a quadratic equation in standard form that has two solutions, 5 and 7 .

Solve each equation. $$ x+7=0 $$

Factor out the GCF from each polynomial. Then factor by grouping. $$ 16 x^{2}+4 x y^{2}+8 x y+2 y^{3} $$

Complete each sentence in your own words. If \(x^{2}+b x+c\) is factorable and \(c\) is positive, then the signs of the last-term factors of the binomials are the same because \(\ldots\)

Factor each trinomial completely. See Examples 1 through 7. \(5 m^{5}+26 m^{3} h^{2}+5 m h^{4}\)

Write an equation that has three solutions, \(0,1,\) and 2

Solve each equation. $$ 3 x+1=0 $$

Multiply. See Section 5.6. \((x-4)(x+4)\)

A compass is accidentally thrown upward and out of an air balloon at a height of 300 feet. The height, \(y\), of the compass at time \(x\) is given by the equation \(y=-16 x^{2}+20 x+300\) a. Find the height of the compass at the given times by filling in the table below. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Time, x (in seconds) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Height, } \mathbf{y} \text { (in feet) } & & & & & & & \\ \hline \end{array} $$ b. Use the table to determine when the compass strikes the ground. c. Use the table to approximate the maximum height of the compass.

Solve each equation. $$ 5 x-15=0 $$

Multiply. $$ (y+3)(y+6) $$

Multiply. See Section 5.6. \((2 x-9)(2 x+9)\)

A rocket is fired upward from the ground with an initial velocity of 100 feet per second. The height, \(y\), of the rocket at any time \(x\) is given by the equation \(y=-16 x^{2}+100 x\) a. Find the height of the rocket at the given times by filling in the table below. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Time, x (in seconds) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Height, } \mathbf{y} \text { (in feet) } & & & & & & & & \\ \hline \end{array} $$ b. Use the table to determine between what two whole-numbered seconds the rocket strikes the ground. c. Use the table to approximate the maximum height of the rocket.

Solve each equation. $$ -2 x=0 $$

Multiply. $$ (b+1)(b-4) $$

An object is thrown upward from the top of an 80 -foot building with an initial velocity of 64 feet per second. Neglecting air resistance, the height of the object after \(t\) seconds is given by \(-16 t^{2}+64 t+80\). Factor this polynomial.

Multiply. See Section 5.6. \((x+2)^{2}\)

Solve each equation. $$ (x-3)(3 x+4)=(x+2)(x-6) $$

Solve each equation. $$ 3 x=0 $$

Multiply. $$ (x-5)(x+10) $$

An object is thrown upward from the top of a 112 -foot building with an initial velocity of 96 feet per second. Neglecting air resistance, the height of the object after \(t\) seconds is given by \(-16 t^{2}+96 t+112 .\) Factor this polynomial.

Multiply. See Section 5.6. \((x+3)^{2}\)

Solve each equation. $$ (2 x-3)(x+6)=(x-9)(x+2) $$

Solve each equation. $$ -5 x+25=0 $$

Fill in the chart by finding two numbers that have the given product and sum. The first column is filled in for you. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline & & \text { 85. } & \text { 86. } & \text { 87. } & \text { 88. } & \text { 89. } & \text { 90. } & \text { 91. } & \text { 92. } \\ \hline \text { Two Numbers } & 4,7 & & & & & & & & \\ \hline \text { Their Product } & 28 & 12 & 20 & 8 & 16 & -10 & -9 & -24 & -36 \\\ \hline \text { Their Sum } & 11 & 8 & 9 & -9 & -10 & 3 & 0 & -5 & -5 \\ \hline \end{array} $$

Factor each trinomial completely. $$ x^{2}+\frac{1}{2} x+\frac{1}{16} $$

Multiply. See Section 5.6. \((2 x-1)^{2}\)

Solve each equation. $$ (2 x-3)(x+8)=(x-6)(x+4) $$

Solve each equation. $$ -4 x-16=0 $$