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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 123. (page 590)

Factor Trinomials of the Form $a x^{2} + b x + c$ using the 鈥榓c鈥� Method.
$16 s^{2} + 40 s + 24$

Short Answer

Expert verified

The solution is $\begin{array}{rcl} 8 (s + 1) (2 s + 3) \end{array}$ .

Step by step solution

01

Step 1. Given information

The given trinomial is $16 s^{2} + 40 s + 24$ .

02

Step 2. Find greatest common factor.

Factor the greatest common factor.

$16 s^{2} + 40 s + 24 = 8 (2 s^{2} + 5 s + 3)$

03

Step 3. Compare the trinomial 2s2+5s+3 to ax2+bx+c and then find the product of ac.

The values are as follows:

$\begin{array}{rcl} a & = & 2 \\ b & = & 5 \\ c & = & 3 \end{array}$

The product is as follows:

$\begin{array}{rcl} a c & = & 2 路 3 \\ = & 6 \end{array}$

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:

$\begin{array}{rcl} 2 路 3 & = & 6 \\ = & a c \\ 2 + 3 & = & 5 \\ = & b \end{array}$

So, two numbers are $2$ and $3$ .

Now, split the middle term of the trinomial $\begin{array}{rcl} 2 s^{2} + 5 s + 3 \end{array}$ as follows:

$\begin{array}{rcl} 2 s^{2} + 5 s + 3 & = & 2 s^{2} + 2 s + 3 s + 3 \end{array}$

05

Step 5. Factor by grouping.

$\begin{array}{rcl} 2 s^{2} + 5 s + 3 & = & 2 s (s + 1) + 3 (+ 1) \\ = & (s + 1) (2 s + 3) \end{array}$

06

Step 6. Check by multiplying all the factors 8(s+1)(2s+3).

$\begin{array}{rcl} 8 (s + 1) (2 s + 3) & = & 8 (2 s^{2} + 2 s + 3 s + 3) \\ = & 16 s^{2} + 40 s + 24 \end{array}$

Hence, the factor is $\begin{array}{rcl} 8 (s + 1) (2 s + 3) \end{array}$ .

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Most popular questions from this chapter

Factor using the 鈥榓c鈥� method:

$4 y^{2} + 8 y + 3$

In the following exercises, factor completely.

$x^{2} + 12 x + 36 - y^{2}$

In the following exercises, factor the greatest common factor from each polynomial.

$12 x^{3} - 10 x$

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error.

$4 w^{2} - 5 w + 1$

In the following exercises, find the greatest common factor.

$12 m^{2} n^{3}, 30 m^{5} n^{3}$

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