Answer :
To factor the expression [tex]\(7x^3 - 42x^2 + 56x\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
- The terms in the expression are [tex]\(7x^3\)[/tex], [tex]\(-42x^2\)[/tex], and [tex]\(56x\)[/tex].
- Each term has a common factor of [tex]\(7x\)[/tex]. So, the GCF is [tex]\(7x\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF [tex]\(7x\)[/tex]:
[tex]\[
7x^3 \div 7x = x^2
\][/tex]
[tex]\[
-42x^2 \div 7x = -6x
\][/tex]
[tex]\[
56x \div 7x = 8
\][/tex]
- After factoring out [tex]\(7x\)[/tex], the expression becomes:
[tex]\[
7x(x^2 - 6x + 8)
\][/tex]
3. Factor the quadratic [tex]\(x^2 - 6x + 8\)[/tex]:
- Look for two numbers that multiply to [tex]\(8\)[/tex] (the constant term) and add up to [tex]\(-6\)[/tex] (the coefficient of the linear term, [tex]\(x\)[/tex]).
- These numbers are [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex], since:
[tex]\[
(-4) \times (-2) = 8
\][/tex]
[tex]\[
(-4) + (-2) = -6
\][/tex]
- Thus, the quadratic can be factored as:
[tex]\[
(x - 4)(x - 2)
\][/tex]
4. Combine everything together:
- The completely factored form of the original expression is:
[tex]\[
7x(x - 4)(x - 2)
\][/tex]
So, the expression [tex]\(7x^3 - 42x^2 + 56x\)[/tex] factors completely to [tex]\(7x(x - 4)(x - 2)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
- The terms in the expression are [tex]\(7x^3\)[/tex], [tex]\(-42x^2\)[/tex], and [tex]\(56x\)[/tex].
- Each term has a common factor of [tex]\(7x\)[/tex]. So, the GCF is [tex]\(7x\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF [tex]\(7x\)[/tex]:
[tex]\[
7x^3 \div 7x = x^2
\][/tex]
[tex]\[
-42x^2 \div 7x = -6x
\][/tex]
[tex]\[
56x \div 7x = 8
\][/tex]
- After factoring out [tex]\(7x\)[/tex], the expression becomes:
[tex]\[
7x(x^2 - 6x + 8)
\][/tex]
3. Factor the quadratic [tex]\(x^2 - 6x + 8\)[/tex]:
- Look for two numbers that multiply to [tex]\(8\)[/tex] (the constant term) and add up to [tex]\(-6\)[/tex] (the coefficient of the linear term, [tex]\(x\)[/tex]).
- These numbers are [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex], since:
[tex]\[
(-4) \times (-2) = 8
\][/tex]
[tex]\[
(-4) + (-2) = -6
\][/tex]
- Thus, the quadratic can be factored as:
[tex]\[
(x - 4)(x - 2)
\][/tex]
4. Combine everything together:
- The completely factored form of the original expression is:
[tex]\[
7x(x - 4)(x - 2)
\][/tex]
So, the expression [tex]\(7x^3 - 42x^2 + 56x\)[/tex] factors completely to [tex]\(7x(x - 4)(x - 2)\)[/tex].
