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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 6: Q 122. (page 590)

Factor Trinomials of the Form $a x^{2} + b x + c$ using the 鈥榓c鈥� Method.
$90 n^{3} + 42 n^{2} - 216 n$

Short Answer

Expert verified

The solution is $\begin{array}{rcl} 6 n (5 n + 9) (3 n - 4) \end{array}$ .

Step by step solution

01

Step 1. Given information

The given trinomial is $90 n^{3} + 42 n^{2} - 216 n$ .

02

Step 2. Find greatest common factor.

Factor the greatest common factor.

$90 n^{3} + 42 n^{2} - 216 n = 6 n (15 n^{2} + 7 n - 36)$

03

Step 3. Compare the trinomial 15n2+7n-36 to ax2+bx+c and then find the product of ac.

The values are as follows:

$\begin{array}{rcl} a & = & 15 \\ b & = & 7 \\ c & = & - 36 \end{array}$

The product is as follows:

$\begin{array}{rcl} a c & = & 15 (- 36) \\ = & - 540 \end{array}$

04

Step 4. Determine two numbers that multiply to ac and add to b.

Two numbers that multiply to $a c$ and add to $b$ are as follows:

localid="1645183790717" $\begin{array}{rcl} (- 20) 27 & = & - 540 \\ = & a c \\ - 20 + 27 & = & 7 \\ = & b \end{array}$

So, two numbers are $- 20$ and localid="1645184184202" $27$ .

Now, split the middle term of the trinomial $15 n^{2} + 7 n - 36$ as follows:

$15 n^{2} + 7 n - 36 = 15 n^{2} + 27 n - 20 n - 36$

05

Step 5. Factor by grouping.

$\begin{array}{rcl} 15 n^{2} + 7 n - 36 & = & 3 n (5 n + 9) - 4 (5 n + 9) \\ = & (5 n + 9) (3 n - 4) \end{array}$

06

Step 6. Check by multiplying all the factors 6n(5n+9)(3n-4).

$\begin{array}{rcl} 6 n (5 n + 9) (3 n - 4) & = & 6 n (15 n^{2} + 27 n - 20 n - 36) \\ = & 6 n (15 n^{2} + 7 n - 36) \\ = & 90 n^{3} 鈥� + 鈥� 42 n^{2} 鈥� 鈭� 鈥� 216 n \end{array}$

Hence, the factor is $\begin{array}{rcl} 6 n (5 n + 9) (3 n - 4) \end{array}$ .

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Most popular questions from this chapter

Factor by Grouping In the following exercises, factor by grouping.

$u v - 9 u + 2 v - 18$

In the following exercises, find the greatest common factor.

$8 a^{2} b^{3}, 10 a b^{2}$

Find the greatest common factors: $25 m^{4}, 35 m^{3}, 20 m^{2}$ .

Factor completely using trial and error:

$56 q^{3} + 320 q^{2} - 96 q$

In the following exercises, factor completely using the difference of squares pattern, if possible.
$m^{2} - 6 m + 9 - 16 n^{2}$

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