Answer :
To solve the expression [tex]\(16x^3 - 48x^2 + 32x\)[/tex], we can factor it step-by-step. Here's how you can do it:
1. Identify the Greatest Common Factor (GCF):
Look at all the terms: [tex]\(16x^3\)[/tex], [tex]\(-48x^2\)[/tex], and [tex]\(32x\)[/tex]. Notice that each of these terms has a common factor. The GCF of the coefficients (16, 48, 32) is 16, and each term also contains at least one [tex]\(x\)[/tex]. Therefore, the GCF of the entire expression is [tex]\(16x\)[/tex].
2. Factor out the GCF:
When you factor out [tex]\(16x\)[/tex] from each term, the expression becomes:
[tex]\[
16x(x^2 - 3x + 2)
\][/tex]
3. Factor the quadratic expression:
Now, we focus on the quadratic expression [tex]\(x^2 - 3x + 2\)[/tex] inside the parentheses.
- We are searching for two numbers that multiply to 2 (the constant term) and add to -3 (the coefficient of [tex]\(x\)[/tex]).
- The numbers that satisfy these conditions are -1 and -2.
4. Write the expression in factored form:
Using the numbers we found, we can factor the quadratic as:
[tex]\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\][/tex]
5. Combine everything together:
Substitute the factored form of the quadratic back into the expression with the GCF:
[tex]\[
16x(x - 1)(x - 2)
\][/tex]
So, the completely factored form of the original expression [tex]\(16x^3 - 48x^2 + 32x\)[/tex] is:
[tex]\[
16x(x - 1)(x - 2)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
Look at all the terms: [tex]\(16x^3\)[/tex], [tex]\(-48x^2\)[/tex], and [tex]\(32x\)[/tex]. Notice that each of these terms has a common factor. The GCF of the coefficients (16, 48, 32) is 16, and each term also contains at least one [tex]\(x\)[/tex]. Therefore, the GCF of the entire expression is [tex]\(16x\)[/tex].
2. Factor out the GCF:
When you factor out [tex]\(16x\)[/tex] from each term, the expression becomes:
[tex]\[
16x(x^2 - 3x + 2)
\][/tex]
3. Factor the quadratic expression:
Now, we focus on the quadratic expression [tex]\(x^2 - 3x + 2\)[/tex] inside the parentheses.
- We are searching for two numbers that multiply to 2 (the constant term) and add to -3 (the coefficient of [tex]\(x\)[/tex]).
- The numbers that satisfy these conditions are -1 and -2.
4. Write the expression in factored form:
Using the numbers we found, we can factor the quadratic as:
[tex]\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\][/tex]
5. Combine everything together:
Substitute the factored form of the quadratic back into the expression with the GCF:
[tex]\[
16x(x - 1)(x - 2)
\][/tex]
So, the completely factored form of the original expression [tex]\(16x^3 - 48x^2 + 32x\)[/tex] is:
[tex]\[
16x(x - 1)(x - 2)
\][/tex]
