Chapter 6

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. $$ x^{2}+x+\frac{1}{4} $$

Multiply. See Section 5.6. \((3 x-5)^{2}\)

Solve each equation. $$ (x+6)(x-6)=(2 x-9)(x+4) $$

Factor each expression completely. $$ x^{2}-\frac{2}{3} x+\frac{1}{9} $$

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. $$ z^{2}(x+1)-3 z(x+1)-70(x+1) $$

See the Concept Check in this section. Do the terms of \(4 x^{2}+19 x+12\) have a common factor (other than 1\() ?\)

Factor each expression completely. $$ x^{2}-\frac{1}{25} $$

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. $$ y^{2}(x+1)-2 y(x+1)-15(x+1) $$

See the Concept Check in this section. Without multiplying, decide which of the following factored forms is not a factored form of \(4 x^{2}+19 x+12\) a. \((2 x+4)(2 x+3)\) b. \((4 x+4)(x+3)\) c. \((4 x+3)(x+4)\) d. \((2 x+2)(2 x+6)\)

Factor each expression completely. $$ (x+2)^{2}-y^{2} $$

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 expression completely. $$ (y-6)^{2}-z^{2} $$

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 expression completely. $$ a^{2}(b-4)-16(b-4) $$

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

Find all positive values of b so that each trinomial is factorable. $$ y^{2}+b y+20 $$

Factor each trinomial completely. \(4 x^{2}+2 x+\frac{1}{4}\)

Factor each expression completely. $$ m^{2}(n+8)-9(n+8) $$

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

Find all positive values of b so that each trinomial is factorable. $$ x^{2}+b x+15 $$

Factor each trinomial completely. \(27 x^{2}+2 x-\frac{1}{9}\)

\(\left(x^{2}+6 x+9\right)-4 y^{2}\) (Hint: Factor the trinomial in parentheses first.)

Which of the following is/are factored form(s) of \(-2 x+14 ?\) a. \(-2(x+7)\) b. \(-2 \cdot x+14\) c. \(-2(x-14)\) d. \(-2(x-7)\)

Factor each trinomial completely. \(4 x^{2}(y-1)^{2}+25 x(y-1)^{2}+25(y-1)^{2}\)

\(\left(x^{2}+2 x+1\right)-36 y^{2}\) (Hint: Factor the trinomial in parentheses first.)

Which of the following is/are factored form(s) of \(8 a-24 ?\) a. \(8 \cdot a-24\) b. \(8(a-3)\) c. \(4(2 a-12)\) d. \(8 \cdot a-2 \cdot 12\)

Factor each trinomial completely. \(3 x^{2}(a+3)^{3}-28 x(a+3)^{3}+25(a+3)^{3}\)

Factor each expression completely. \(x^{2 n}-100\)

Which of the following expressions are factored? $$ (a+6)(a+2) $$

Factor each expression completely. $$ x^{2 n}-81 $$

Which of the following expressions are factored? $$ (x+5)(x+y) $$

Fill in the blank so that \(x^{2}+_______x+16\) is a perfect square trinomial.

Which of the following expressions are factored? $$ 5(2 y+z)-b(2 y+z) $$

Find all positive values of \(c\) so that each trinomial is factorable. \(5 x^{2}+7 x+c\)

Fill in the blank so that \(9 x^{2}+_________x+25\) is a perfect square trinomial.

Which of the following expressions are factored? $$ 3 x(a+2 b)+2(a+2 b) $$

Find all positive values of \(c\) so that each trinomial is factorable. \(3 x^{2}-8 x+c\)

Describe a perfect square trinomial.

The annual cotton crop yield (in 1000 bales) in the United States for the period \(2003-2007\) can be approximated by the polynomial \(-1264 x^{2}+5056 x+18,960,\) where \(x\) is the number of years after 2003. (Source: Based on data from the National Agricultural Statistics Service) a. Find the approximate amount of the cotton harvest in 2004. To do so, let \(x=1\) and evaluate \(-1264 x^{2}+5056 x+18,960\) b. Find the approximate amount of cotton harvested in 2007 . c. Factor the polynomial \(-1264 x^{2}+5056 x+18,960\).

Write a perfect square trinomial that factors as \((x+3 y)^{2}\)

The polynomial \(-30 x^{2}+180 x+210\) represents the approximate number of visitors (in thousands) per year to the White House during \(2003-2007\). In this polynomial, \(x\) represents the years since 2003 . (Source: Based on data from the National Park Service) a. Find the approximate number of visitors to the White House in \(2005 .\) To do so, let \(x=2\) and evaluate \(-30 x^{2}+180 x+210\). b. Find the approximate number of visitors to the White House in 2006 . c. Factor out the GCF from the polynomial \(-30 x^{2}+180 x+210\)

Find all positive values of \(c\) so that each trinomial is factorable. A student in your class factored \(6 x^{2}+7 x+1\) as \((3 x+1)(2 x+1)\). Write down how you would explain the student's error.

What binomial multiplied by \((x-6)\) gives the difference of two squares?

What binomial multiplied by \((5+y)\) gives the difference of two squares?

The area of the largest square in the figure is \((a+b)^{2}\). What factoring formula from this section is visually represented by this square?

The Toroweap Overlook, on the North Rim of the Grand Canyon, lies 3000 vertical feet above the Colorado River. The view is spectacular, and the sheer drop is dramatic. A film crew creating a documentary about the Grand Canyon has suspended a camera platform 296 feet below the Overlook. A camera filter comes loose and falls to the river below. The height of the filter above the river after \(t\) seconds is given by the expression \(2704-16 t^{2}\). a. Find the height of the filter above the river after 3 seconds. b. Find the height of the filter above the river after 7 seconds. c. To the nearest whole second, estimate when the filter lands in the river. d. Factor \(2704-16 t^{2}\).

Construct a binomial whose greatest common factor is \(5 a^{3}\). (Hint: Multiply \(5 a^{3}\) by a binomial whose terms contain no common factor other than \(\left.1: 5 a^{3}(\square+\square) .\right)\)