Answer :
To simplify the expression [tex]\(16x^3 - 48x^2 + 32x\)[/tex], we can follow these steps:
1. Factor Out the Greatest Common Factor:
First, look for the greatest common factor (GCF) in the expression. We can see that each term in the expression shares the factor [tex]\(16x\)[/tex]. Therefore, we can factor out [tex]\(16x\)[/tex] from the expression:
[tex]\[
16x^3 - 48x^2 + 32x = 16x(x^2 - 3x + 2)
\][/tex]
2. Simplify the Expression Inside the Parenthesis:
Next, take the quadratic expression inside the parentheses, [tex]\(x^2 - 3x + 2\)[/tex], and factor it. To factor the quadratic, we need to find two numbers that multiply to the constant term (2) and add to the coefficient of the linear term (-3).
The numbers that satisfy this are -1 and -2, since [tex]\((-1) \times (-2) = 2\)[/tex] and [tex]\((-1) + (-2) = -3\)[/tex]. So, we can write:
[tex]\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\][/tex]
3. Combine the Factors:
Now, substitute the factored form of the quadratic back into the overall expression. This gives:
[tex]\[
16x(x^2 - 3x + 2) = 16x(x - 1)(x - 2)
\][/tex]
Thus, the fully factored form of the expression [tex]\(16x^3 - 48x^2 + 32x\)[/tex] is [tex]\(16x(x - 1)(x - 2)\)[/tex].
1. Factor Out the Greatest Common Factor:
First, look for the greatest common factor (GCF) in the expression. We can see that each term in the expression shares the factor [tex]\(16x\)[/tex]. Therefore, we can factor out [tex]\(16x\)[/tex] from the expression:
[tex]\[
16x^3 - 48x^2 + 32x = 16x(x^2 - 3x + 2)
\][/tex]
2. Simplify the Expression Inside the Parenthesis:
Next, take the quadratic expression inside the parentheses, [tex]\(x^2 - 3x + 2\)[/tex], and factor it. To factor the quadratic, we need to find two numbers that multiply to the constant term (2) and add to the coefficient of the linear term (-3).
The numbers that satisfy this are -1 and -2, since [tex]\((-1) \times (-2) = 2\)[/tex] and [tex]\((-1) + (-2) = -3\)[/tex]. So, we can write:
[tex]\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\][/tex]
3. Combine the Factors:
Now, substitute the factored form of the quadratic back into the overall expression. This gives:
[tex]\[
16x(x^2 - 3x + 2) = 16x(x - 1)(x - 2)
\][/tex]
Thus, the fully factored form of the expression [tex]\(16x^3 - 48x^2 + 32x\)[/tex] is [tex]\(16x(x - 1)(x - 2)\)[/tex].
