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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 121. (page 590)

Factor Trinomials of the Form $a x^{2} + b x + c$ using the 鈥榓c鈥� Method.
$48 z^{3} 鈭� 102 z^{2} 鈭� 45 z$

Short Answer

Expert verified

The solution is $\begin{array}{rcl} 3 z (2 z - 5) (8 z + 3) \end{array}$ .

Step by step solution

01

Step 1. Given information

The given trinomial is $48 z^{3} 鈭� 102 z^{2} 鈭� 45 z$ .

02

Step 2. Find greatest common factor.

Factor the greatest common factor.

$48 z^{3} 鈭� 102 z^{2} 鈭� 45 z = 3 z (16 z^{2} - 34 z - 15)$

03

Step 3. Compare the trinomial 16z2-34z-15 to ax2+bx+c and then find the product of ac.

The values are as follows:

$\begin{array}{rcl} a & = & 16 \\ b & = & - 34 \\ c & = & - 15 \end{array}$

The product is as follows:

$\begin{array}{rcl} a c & = & 16 (- 15) \\ = & - 240 \end{array}$

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:

$\begin{array}{rcl} - 40 (6) & = & - 240 \\ = & a c \\ - 40 + 6 & = & - 34 \\ = & b \end{array}$

So, two numbers are $- 40$ and $6$ .

Now, split the middle term of the trinomial $16 z^{2} - 34 z - 15$ as follows:

$\begin{array}{rcl} 16 z^{2} - 34 z - 15 & = & 16 z^{2} - 40 z + 6 z - 15 \end{array}$

05

Step 5. Factor by grouping.

$\begin{array}{rcl} 16 z^{2} - 34 z - 15 & = & 8 z (2 z - 5) + 3 (2 z - 5) \\ = & (2 z - 5) (8 z + 3) \end{array}$

06

Step 6. Check by multiplying all the factors 3z(2z-5)(8z+3).

$\begin{array}{rcl} 3 z (2 z - 5) (8 z + 3) & = & 3 z (16 z^{2} - 40 z + 6 z - 15) \\ = & 3 z (16 z^{2} - 34 z - 15) \\ = & 48 z^{3} - 102 z - 45 z \end{array}$

Hence, the factor is $\begin{array}{rcl} 3 z (2 z - 5) (8 z + 3) \end{array}$ .

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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, factor the greatest common factor from each polynomial.

$10 q^{2} + 14 q + 20$

In the following exercises, find the greatest common factor.

$27 p^{2} q^{3}, 45 p^{3} q^{4}, 9 p^{4} q^{3}$

In the following exercises, factor completely using the difference of squares pattern, if possible.
$x^{2} - 16 x + 64 - y^{2}$

In the following exercises, factor completely.

$48 q^{3} - 24 q^{2} + 3 q$

See all solutions

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