Answer :
To factor the expression [tex]\(18x^3 + 25x^2 - 3x\)[/tex], we can follow these steps:
1. Identify a Common Factor:
Start by identifying the greatest common factor (GCF) of the terms in the polynomial. In [tex]\(18x^3 + 25x^2 - 3x\)[/tex], all terms share a common factor of [tex]\(x\)[/tex]. Factor out [tex]\(x\)[/tex]:
[tex]\[
x(18x^2 + 25x - 3)
\][/tex]
2. Factor the Quadratic:
Now, focus on factoring the quadratic expression [tex]\(18x^2 + 25x - 3\)[/tex]. We need to find two numbers whose product is [tex]\(18 \times (-3) = -54\)[/tex] and whose sum is [tex]\(25\)[/tex].
The pair of numbers that work are [tex]\(27\)[/tex] and [tex]\(-2\)[/tex] because [tex]\(27 \times (-2) = -54\)[/tex] and [tex]\(27 + (-2) = 25\)[/tex].
3. Rewrite and Group:
Rewrite the middle term [tex]\(25x\)[/tex] using the numbers we found:
[tex]\[
18x^2 + 27x - 2x - 3
\][/tex]
Group the terms to factor by grouping:
[tex]\[
(18x^2 + 27x) + (-2x - 3)
\][/tex]
4. Factor by Grouping:
Factor out the common factors in each group:
- From [tex]\(18x^2 + 27x\)[/tex], factor out [tex]\(9x\)[/tex]:
[tex]\[
9x(2x + 3)
\][/tex]
- From [tex]\(-2x - 3\)[/tex], factor out [tex]\(-1\)[/tex]:
[tex]\[
-1(2x + 3)
\][/tex]
5. Combine the Factored Terms:
Now, notice that both groups contain a common binomial factor [tex]\((2x + 3)\)[/tex]:
[tex]\[
(9x - 1)(2x + 3)
\][/tex]
6. Combine All Parts:
The factored expression of the entire original expression is:
[tex]\[
x(2x + 3)(9x - 1)
\][/tex]
So, the fully factored form of the polynomial [tex]\(18x^3 + 25x^2 - 3x\)[/tex] is [tex]\(x(2x + 3)(9x - 1)\)[/tex].
1. Identify a Common Factor:
Start by identifying the greatest common factor (GCF) of the terms in the polynomial. In [tex]\(18x^3 + 25x^2 - 3x\)[/tex], all terms share a common factor of [tex]\(x\)[/tex]. Factor out [tex]\(x\)[/tex]:
[tex]\[
x(18x^2 + 25x - 3)
\][/tex]
2. Factor the Quadratic:
Now, focus on factoring the quadratic expression [tex]\(18x^2 + 25x - 3\)[/tex]. We need to find two numbers whose product is [tex]\(18 \times (-3) = -54\)[/tex] and whose sum is [tex]\(25\)[/tex].
The pair of numbers that work are [tex]\(27\)[/tex] and [tex]\(-2\)[/tex] because [tex]\(27 \times (-2) = -54\)[/tex] and [tex]\(27 + (-2) = 25\)[/tex].
3. Rewrite and Group:
Rewrite the middle term [tex]\(25x\)[/tex] using the numbers we found:
[tex]\[
18x^2 + 27x - 2x - 3
\][/tex]
Group the terms to factor by grouping:
[tex]\[
(18x^2 + 27x) + (-2x - 3)
\][/tex]
4. Factor by Grouping:
Factor out the common factors in each group:
- From [tex]\(18x^2 + 27x\)[/tex], factor out [tex]\(9x\)[/tex]:
[tex]\[
9x(2x + 3)
\][/tex]
- From [tex]\(-2x - 3\)[/tex], factor out [tex]\(-1\)[/tex]:
[tex]\[
-1(2x + 3)
\][/tex]
5. Combine the Factored Terms:
Now, notice that both groups contain a common binomial factor [tex]\((2x + 3)\)[/tex]:
[tex]\[
(9x - 1)(2x + 3)
\][/tex]
6. Combine All Parts:
The factored expression of the entire original expression is:
[tex]\[
x(2x + 3)(9x - 1)
\][/tex]
So, the fully factored form of the polynomial [tex]\(18x^3 + 25x^2 - 3x\)[/tex] is [tex]\(x(2x + 3)(9x - 1)\)[/tex].
