# 6.7 Factoring trinomials with leading coefficient other than 1  (Page 3/3)

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## Sample set b

Factor $24{x}^{2}-41x+12$ .

Factor the first and last terms.

 $\text{\hspace{0.17em}}24{x}^{2}$ 12 $24x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}x$ $-12,\text{\hspace{0.17em}}\text{\hspace{0.17em}}-1$ $12x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}2x$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}-6,\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}8x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}3x$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4,\text{\hspace{0.17em}}\text{\hspace{0.17em}}-3$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}6x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}4x$

Rather than starting with the $24x,x$ and $-12,-1$ , pick some intermediate values, $8x$ and $3x$ , the $6x$ and $4x$ , or the $-6$ and $-2$ , or the $-4$ and $-3$ .

$24{x}^{2}-41x+12=\left(8x-3\right)\left(3x-4\right)$

## Practice set b

Factor $48{x}^{2}+22x-15$ .

$\left(6x+5\right)\left(8x-3\right)$

Factor $54{y}^{2}+39yw-28{w}^{2}$ .

$\left(9y-4w\right)\left(6y+7w\right)$

## The collect and discard method of factoring $a{x}^{2}+bx+c$

Consider the polynomial $6{x}^{2}+x-12$ . We begin by identifying $a$ and $c$ . In this case, $a=6$ and $c=-12$ . We start out as we would with $a=1$ .

$6{x}^{2}+x-12:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\begin{array}{ll}6x\hfill & \hfill \end{array}\right)\left(\begin{array}{ll}6x\hfill & \hfill \end{array}\right)$

Now, compute $a\cdot c$ .

$a\cdot c=\left(6\right)\left(-12\right)=-72$

Find the factors of $-72$ that add to 1, the coefficient of $x$ , the linear term. The factors are 9 and $-8$ . Include these factors in the parentheses.

$6{x}^{2}+x-12:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(6x+9\right)\left(6x-8\right)$

But we have included too much. We must eliminate the surplus. Factor each parentheses.

$6{x}^{2}+x-12:\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\left(2x+3\right)\cdot 2\left(3x-4\right)$

Discard the factors that multiply to $a=6$ . In this case, 3 and 2. We are left with the proper factorization.

$6{x}^{2}+x-12=\left(2x+3\right)\left(3x-4\right)$

## Sample set c

Factor $10{x}^{2}+23x-5$ .

Identify $a=10$ and $b=-5$ .

$10{x}^{2}+23x-5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\begin{array}{ll}10x\hfill & \hfill \end{array}\right)\left(\begin{array}{ll}10x\hfill & \hfill \end{array}\right)$

Compute

$a\cdot c=\left(10\right)\left(-5\right)=-50$

Find the factors of $-50$ that add to $+23$ , the coefficient of $x$ , the linear term. The factors are 25 and $-2$ . Place these numbers into the parentheses.

$10{x}^{2}+23x-5:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(10x+25\right)\left(10x-2\right)$

We have collected too much. Factor each set of parentheses and eliminate the surplus.

$10{x}^{2}+23x-5:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(5\right)\left(2x+5\right)\cdot \left(2\right)\left(5x-1\right)$

Discard the factors that multiply to $a=10$ . In this case, 5 and 2.

$10{x}^{2}+23x-5=\left(2x+5\right)\left(5x-1\right)$

Factor $8{x}^{2}-30x-27$ .

Identify $a=8$ and $c=-27$ .

$8{x}^{2}-30x-27:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\begin{array}{ll}8x\hfill & \hfill \end{array}\right)\left(\begin{array}{ll}8x\hfill & \hfill \end{array}\right)$

Compute

$a\cdot c=\left(8\right)\left(-27\right)=-216$

Find the factors of $-216$ that add to $-30$ , the coefficient of $x$ , the linear term. This requires some thought. The factors are $-36$ and 6. Place these numbers into the parentheses.

$8{x}^{2}-30x-27:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(8x-36\right)\left(8x+6\right)$

We have collected too much. Factor each set of parentheses and eliminate the surplus.

$8{x}^{2}-30x-27:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(4\right)\left(2x-9\right)\cdot \left(2\right)\left(4x+3\right)$

Discard the factors that multiply to $a=8$ . In this case, 4 and 2.

$8{x}^{2}-30x-27=\left(2x-9\right)\left(4x+3\right)$

Factor $18{x}^{2}-5xy-2{y}^{2}$ .

Identify $a=18$ and $c=-2$ .

$18{x}^{2}-5xy-2{y}^{2}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\begin{array}{ll}18x\hfill & \hfill \end{array}\right)\left(\begin{array}{ll}18x\hfill & \hfill \end{array}\right)$

Compute

$a\cdot c=\left(18\right)\left(-2\right)=-36$

Find the factors of $-36$ that add to $-5$ , the coefficient of $xy$ . In this case, $-9$ and 4. Place these numbers into the parentheses, affixing $y$ to each.

$18{x}^{2}-5xy-2{y}^{2}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(18x-9y\right)\left(18x+4y\right)$

We have collected too much. Factor each set of parentheses and eliminate the surplus.

$18{x}^{2}-5xy-2{y}^{2}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(9\right)\left(2x-y\right)\cdot \left(2\right)\left(9x+2y\right)$

Discard the factors that multiply to $a=18$ . In this case, 9 and 4.

$18{x}^{2}-5xy-2{y}^{2}=\left(2x-y\right)\left(9x+2y\right)$

## Practice set c

Factor $6{x}^{2}+7x-3$ .

$\left(3x-1\right)\left(2x+3\right)$

Factor $14{x}^{2}-31x-10$ .

$\left(7x+2\right)\left(2x-5\right)$

Factor $48{x}^{2}+22x-15$ .

$\left(6x+5\right)\left(8x-3\right)$

Factor $10{x}^{2}-23xw+12{w}^{2}$ .

$\left(5x-4w\right)\left(2x-3w\right)$

## Exercises

Factor the following problems, if possible.

${x}^{2}+3x+2$

$\left(x+2\right)\left(x+1\right)$

${x}^{2}+7x+12$

$2{x}^{2}+7x+5$

$\left(2x+5\right)\left(x+1\right)$

$3{x}^{2}+4x+1$

$2{x}^{2}+11x+12$

$\left(2x+3\right)\left(x+4\right)$

$10{x}^{2}+33x+20$

$3{x}^{2}-x-4$

$\left(3x-4\right)\left(x+1\right)$

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

$4{x}^{2}+8x-21$

$\left(2x-3\right)\left(2x+7\right)$

$2{a}^{2}-a-3$

$9{a}^{2}-7a+2$

not factorable

$16{a}^{2}+16a+3$

$16{y}^{2}-26y+3$

$\left(8y-1\right)\left(2y-3\right)$

$3{y}^{2}+14y-5$

$10{x}^{2}+29x+10$

$\left(5x+2\right)\left(2x+5\right)$

$14{y}^{2}+29y-15$

$81{a}^{2}+19a+2$

not factorable

$24{x}^{2}+34x+5$

$24{x}^{2}-34x+5$

$\left(6x-1\right)\left(4x-5\right)$

$24{x}^{2}-26x-5$

$24{x}^{2}+26x-5$

$\left(6x-1\right)\left(4x+5\right)$

$6{a}^{2}+13a+6$

$6{x}^{2}+5xy+{y}^{2}$

$\left(3x+y\right)\left(2x+y\right)$

$6{a}^{2}-ay-{y}^{2}$

For the following problems, the given trinomial occurs when solving the corresponding applied problem. Factor each trinomial. You do not need to solve the problem.

$5{r}^{2}-24r-5$ .

It takes 5 hours to paddle a boat 12 miles downstream and then back. The current flows at the rate of 1 mile per hour. At what rate was the boat paddled?

$\left(5r+1\right)\left(r-5\right)$

${x}^{2}+5x-84$ .

The length of a rectangle is 5 inches more than the width of the rectangle. If the area of the rectangle is 84 square inches, what are the length and width of the rectangle?

${x}^{2}+24x-145$ .

A square measures 12 inches on each side. Another square is to be drawn around this square in such a way that the total area is 289 square inches. What is the distance from the edge of the smaller square to the edge of the larger square? (The two squares have the same center.)

$\left(x+29\right)\left(x-5\right)$

${x}^{2}+8x-20$ .

A woman wishes to construct a rectangular box that is open at the top. She wishes it to be 4 inches high and have a rectangular base whose length is three times the width. The material used for the base costs $2 per square inch, and the material used for the sides costs$1.50 per square inch. The woman will spend exactly \$120 for materials. Find the dimension of the box (length of the base, width of the base, and height).

For the following problems, factor the trinomials if possible.

$16{x}^{2}-8xy-3{y}^{2}$

$\left(4x+y\right)\left(4x-3y\right)$

$6{a}^{2}+7ab+2{b}^{2}$

$12{a}^{2}+7ab+12{b}^{2}$

not factorable

$9{x}^{2}+18xy+8{y}^{2}$

$8{a}^{2}+10ab-6{b}^{2}$

$2\left(4{a}^{2}+5ab-3{b}^{2}\right)$

$12{a}^{2}+54a-90$

$12{b}^{4}+30{b}^{2}a+12{a}^{2}$

$6\left(2{b}^{2}+a\right)\left({b}^{2}+2a\right)$

$30{a}^{4}{b}^{4}-3{a}^{2}{b}^{2}-6{c}^{2}$

$3{a}^{6}-3{a}^{3}{b}^{2}-18{b}^{4}$

$3\left({a}^{3}+2{b}^{2}\right)\left({a}^{3}-3{b}^{2}\right)$

$20{a}^{2}{b}^{2}+2ab{c}^{2}-6{a}^{2}{c}^{4}$

$14{a}^{2}{z}^{2}-40{a}^{3}{z}^{2}-46{a}^{4}{z}^{2}$

$2{a}^{2}{z}^{2}\left(7-20a-23{a}^{2}\right)\text{\hspace{0.17em}or\hspace{0.17em}}-2{a}^{2}{z}^{2}\left(23{a}^{2}+20a-7\right)$

## Exercises for review

( [link] ) Simplify ${\left({a}^{3}{b}^{6}\right)}^{4}$ .

( [link] ) Find the product. ${x}^{2}\left(x-3\right)\left(x+4\right)$ .

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

( [link] ) Find the product. ${\left(5m-3n\right)}^{2}$ .

( [link] ) Solve the equation $5\left(2x-1\right)-4\left(x+7\right)=0$ .

$x=\frac{11}{2}$

( [link] ) Factor ${x}^{5}-8{x}^{4}+7{x}^{3}$ .

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