Answer :
To solve the equation [tex]\( x^3 + x^2 - 42x = 0 \)[/tex] by factoring, follow these steps:
1. Factor out the Greatest Common Factor (GCF):
Notice that each term in the equation has a common factor, which is [tex]\( x \)[/tex]. So, we can factor [tex]\( x \)[/tex] out of the equation:
[tex]\[
x(x^2 + x - 42) = 0
\][/tex]
2. Factor the Quadratic Expression:
Next, we need to factor the quadratic expression [tex]\( x^2 + x - 42 \)[/tex]. We are looking for two numbers that multiply to [tex]\(-42\)[/tex] (the constant term) and add up to [tex]\(1\)[/tex] (the coefficient of the middle term). The numbers [tex]\(-6\)[/tex] and [tex]\(7\)[/tex] satisfy these conditions because:
[tex]\[
-6 \times 7 = -42 \quad \text{and} \quad -6 + 7 = 1
\][/tex]
Using these numbers, we can factor the quadratic expression:
[tex]\[
x^2 + x - 42 = (x + 7)(x - 6)
\][/tex]
3. Write the Fully Factored Equation:
Substitute the factored quadratic expression back into the original equation:
[tex]\[
x(x + 7)(x - 6) = 0
\][/tex]
4. Find the Solutions:
Set each factor in the equation equal to zero and solve for [tex]\( x \)[/tex]:
- [tex]\( x = 0 \)[/tex]
- [tex]\( x + 7 = 0 \)[/tex] which gives [tex]\( x = -7 \)[/tex]
- [tex]\( x - 6 = 0 \)[/tex] which gives [tex]\( x = 6 \)[/tex]
The zeros of the equation are [tex]\( x = 0 \)[/tex], [tex]\( x = -7 \)[/tex], and [tex]\( x = 6 \)[/tex].
5. Check the Options:
- E: 0
- G: -7
- C: 6
These are the zeros that are solutions to the equation, so the correct options are E (0), G (-7), and C (6).
1. Factor out the Greatest Common Factor (GCF):
Notice that each term in the equation has a common factor, which is [tex]\( x \)[/tex]. So, we can factor [tex]\( x \)[/tex] out of the equation:
[tex]\[
x(x^2 + x - 42) = 0
\][/tex]
2. Factor the Quadratic Expression:
Next, we need to factor the quadratic expression [tex]\( x^2 + x - 42 \)[/tex]. We are looking for two numbers that multiply to [tex]\(-42\)[/tex] (the constant term) and add up to [tex]\(1\)[/tex] (the coefficient of the middle term). The numbers [tex]\(-6\)[/tex] and [tex]\(7\)[/tex] satisfy these conditions because:
[tex]\[
-6 \times 7 = -42 \quad \text{and} \quad -6 + 7 = 1
\][/tex]
Using these numbers, we can factor the quadratic expression:
[tex]\[
x^2 + x - 42 = (x + 7)(x - 6)
\][/tex]
3. Write the Fully Factored Equation:
Substitute the factored quadratic expression back into the original equation:
[tex]\[
x(x + 7)(x - 6) = 0
\][/tex]
4. Find the Solutions:
Set each factor in the equation equal to zero and solve for [tex]\( x \)[/tex]:
- [tex]\( x = 0 \)[/tex]
- [tex]\( x + 7 = 0 \)[/tex] which gives [tex]\( x = -7 \)[/tex]
- [tex]\( x - 6 = 0 \)[/tex] which gives [tex]\( x = 6 \)[/tex]
The zeros of the equation are [tex]\( x = 0 \)[/tex], [tex]\( x = -7 \)[/tex], and [tex]\( x = 6 \)[/tex].
5. Check the Options:
- E: 0
- G: -7
- C: 6
These are the zeros that are solutions to the equation, so the correct options are E (0), G (-7), and C (6).
