Question

complete the factorization of  x cubed plus 2 x squared minus 63 x including the greatest common factor.

Answer 1

To factor the expression \( x^3 + 2x^2 - 63x \), we can start by identifying the greatest common factor (GCF).

1. **Find the GCF**: The GCF of the terms \( x^3 \), \( 2x^2 \), and \( -63x \) is \( x \).

2. **Factor out the GCF**: We can factor out \( x \) from the entire expression:
\[
x^3 + 2x^2 - 63x = x(x^2 + 2x - 63)
\]

3. **Factor the quadratic**: Next, we need to factor the quadratic \( x^2 + 2x - 63 \). We will look for two numbers that multiply to \(-63\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers \(9\) and \(-7\) work because:
\[
9 \times (-7) = -63 \quad \text{and} \quad 9 + (-7) = 2
\]

Therefore, we can factor the quadratic as:
\[
x^2 + 2x - 63 = (x + 9)(x - 7)
\]

4. **Combine everything**: Now we can write the complete factorization of the original expression:
\[
x^3 + 2x^2 - 63x = x(x + 9)(x - 7)
\]

Thus, the complete factorization including the greatest common factor is:
\[
\boxed{x(x + 9)(x - 7)}
\]