Answer :
To factor the expression [tex]\( 36x^3 - 48x^2 - 9x \)[/tex], let's go through the process step-by-step:
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of the terms in the expression. The terms are [tex]\( 36x^3 \)[/tex], [tex]\( -48x^2 \)[/tex], and [tex]\( -9x \)[/tex].
- The coefficients are 36, -48, and -9. The greatest common factor of these numbers is 3.
- Each term also has at least one factor of [tex]\( x \)[/tex]. So, the GCF of the expression is [tex]\( 3x \)[/tex].
2. Factor out the GCF:
Factor [tex]\( 3x \)[/tex] out of each term:
[tex]\[
36x^3 - 48x^2 - 9x = 3x(12x^2 - 16x - 3)
\][/tex]
3. Factor the Quadratic:
Next, we need to factor the quadratic expression inside the parentheses: [tex]\( 12x^2 - 16x - 3 \)[/tex].
To do this, we can look for two numbers that multiply to [tex]\( 12 \times -3 = -36 \)[/tex] and add to [tex]\(-16\)[/tex].
The numbers [tex]\(-18\)[/tex] and [tex]\(2\)[/tex] fit these requirements because:
[tex]\[
-18 \times 2 = -36 \quad \text{and} \quad -18 + 2 = -16
\][/tex]
Now, rewrite the quadratic by splitting the middle term using these numbers:
[tex]\[
12x^2 - 18x + 2x - 3
\][/tex]
Now, group the terms and factor by grouping:
[tex]\[
(12x^2 - 18x) + (2x - 3)
\][/tex]
Factor out the common factor from each group:
[tex]\[
6x(2x - 3) + 1(2x - 3)
\][/tex]
Notice that [tex]\(2x - 3\)[/tex] is a common factor:
[tex]\[
(6x + 1)(2x - 3)
\][/tex]
4. Combine Everything:
Now, combine everything back with the factored GCF:
[tex]\[
3x(6x + 1)(2x - 3)
\][/tex]
Therefore, the expression [tex]\( 36x^3 - 48x^2 - 9x \)[/tex] factors to [tex]\( 3x(6x + 1)(2x - 3) \)[/tex].
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of the terms in the expression. The terms are [tex]\( 36x^3 \)[/tex], [tex]\( -48x^2 \)[/tex], and [tex]\( -9x \)[/tex].
- The coefficients are 36, -48, and -9. The greatest common factor of these numbers is 3.
- Each term also has at least one factor of [tex]\( x \)[/tex]. So, the GCF of the expression is [tex]\( 3x \)[/tex].
2. Factor out the GCF:
Factor [tex]\( 3x \)[/tex] out of each term:
[tex]\[
36x^3 - 48x^2 - 9x = 3x(12x^2 - 16x - 3)
\][/tex]
3. Factor the Quadratic:
Next, we need to factor the quadratic expression inside the parentheses: [tex]\( 12x^2 - 16x - 3 \)[/tex].
To do this, we can look for two numbers that multiply to [tex]\( 12 \times -3 = -36 \)[/tex] and add to [tex]\(-16\)[/tex].
The numbers [tex]\(-18\)[/tex] and [tex]\(2\)[/tex] fit these requirements because:
[tex]\[
-18 \times 2 = -36 \quad \text{and} \quad -18 + 2 = -16
\][/tex]
Now, rewrite the quadratic by splitting the middle term using these numbers:
[tex]\[
12x^2 - 18x + 2x - 3
\][/tex]
Now, group the terms and factor by grouping:
[tex]\[
(12x^2 - 18x) + (2x - 3)
\][/tex]
Factor out the common factor from each group:
[tex]\[
6x(2x - 3) + 1(2x - 3)
\][/tex]
Notice that [tex]\(2x - 3\)[/tex] is a common factor:
[tex]\[
(6x + 1)(2x - 3)
\][/tex]
4. Combine Everything:
Now, combine everything back with the factored GCF:
[tex]\[
3x(6x + 1)(2x - 3)
\][/tex]
Therefore, the expression [tex]\( 36x^3 - 48x^2 - 9x \)[/tex] factors to [tex]\( 3x(6x + 1)(2x - 3) \)[/tex].
