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Answer :
To factor the given cubic expression [tex]\(3x^3 - 48x^2 + 180x\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the GCF of all the terms in the expression. The terms are [tex]\(3x^3\)[/tex], [tex]\(-48x^2\)[/tex], and [tex]\(180x\)[/tex].
The GCF is [tex]\(3x\)[/tex]. This is because:
- Each term has at least one factor of [tex]\(x\)[/tex].
- The coefficients 3, 48, and 180 have a common factor of 3.
2. Factor Out the GCF:
Factor [tex]\(3x\)[/tex] out of each term in the expression:
[tex]\[
3x^3 - 48x^2 + 180x = 3x(x^2 - 16x + 60)
\][/tex]
3. Factor the Quadratic Expression:
Now, focus on the quadratic expression inside the parentheses: [tex]\(x^2 - 16x + 60\)[/tex].
We need to find two numbers that multiply to 60 and add up to -16.
The numbers that satisfy this are -10 and -6 because:
- [tex]\(-10 \times -6 = 60\)[/tex]
- [tex]\(-10 + (-6) = -16\)[/tex]
Using these numbers, factor the quadratic:
[tex]\[
x^2 - 16x + 60 = (x - 10)(x - 6)
\][/tex]
4. Write the Fully Factored Form:
Substitute the factored quadratic back into the expression:
[tex]\[
3x(x^2 - 16x + 60) = 3x(x - 10)(x - 6)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(3x^3 - 48x^2 + 180x\)[/tex] is:
[tex]\[
3x(x - 10)(x - 6)
\][/tex]
This is the final answer.
1. Identify the Greatest Common Factor (GCF):
First, look for the GCF of all the terms in the expression. The terms are [tex]\(3x^3\)[/tex], [tex]\(-48x^2\)[/tex], and [tex]\(180x\)[/tex].
The GCF is [tex]\(3x\)[/tex]. This is because:
- Each term has at least one factor of [tex]\(x\)[/tex].
- The coefficients 3, 48, and 180 have a common factor of 3.
2. Factor Out the GCF:
Factor [tex]\(3x\)[/tex] out of each term in the expression:
[tex]\[
3x^3 - 48x^2 + 180x = 3x(x^2 - 16x + 60)
\][/tex]
3. Factor the Quadratic Expression:
Now, focus on the quadratic expression inside the parentheses: [tex]\(x^2 - 16x + 60\)[/tex].
We need to find two numbers that multiply to 60 and add up to -16.
The numbers that satisfy this are -10 and -6 because:
- [tex]\(-10 \times -6 = 60\)[/tex]
- [tex]\(-10 + (-6) = -16\)[/tex]
Using these numbers, factor the quadratic:
[tex]\[
x^2 - 16x + 60 = (x - 10)(x - 6)
\][/tex]
4. Write the Fully Factored Form:
Substitute the factored quadratic back into the expression:
[tex]\[
3x(x^2 - 16x + 60) = 3x(x - 10)(x - 6)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(3x^3 - 48x^2 + 180x\)[/tex] is:
[tex]\[
3x(x - 10)(x - 6)
\][/tex]
This is the final answer.
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