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: \[ \boxed{x(x + 9)(x - 7)} \]