Chapter 6: Q 122. (page 590)

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ 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

Step 1. Given information

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

Step 2. Find greatest common factor.

Factor the greatest common factor.

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

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:

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

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:

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

Step 5. Factor by grouping.

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

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

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

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

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find greatest common factor.

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

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

Step 5. Factor by grouping.

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

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Factor Trinomials of the Form ax2+bx+cusing the ‘ac’ Method.90n3+42n2-216n

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find greatest common factor.

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

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

Step 5. Factor by grouping.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Pure Maths

Probability and Statistics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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