Problem 8: \( 18 x^{3}+9 x^{2} \) (first taught in lesson 27)

Certainly! Let's work through **Problem 8** step by step. ### Problem 8: Factor the expression \( 18x^{3} + 9x^{2} \). ### Step 1: Identify the Greatest Common Factor (GCF) First, determine the GCF of the coefficients and the variables in each term. - **Coefficients:** The coefficients are 18 and 9. The GCF of 18 and 9 is **9**. - **Variables:** Both terms have \( x^{3} \) and \( x^{2} \). The smallest power of \( x \) is \( x^{2} \). So, the GCF of the entire expression is **\( 9x^{2} \)**. ### Step 2: Factor Out the GCF Divide each term by the GCF and place the GCF outside the parentheses. \[ 18x^{3} + 9x^{2} = 9x^{2} \left( \frac{18x^{3}}{9x^{2}} + \frac{9x^{2}}{9x^{2}} \right) \] Simplify inside the parentheses: \[ = 9x^{2} (2x + 1) \] ### Final Factored Form \[ 18x^{3} + 9x^{2} = 9x^{2}(2x + 1) \] ### Verification To ensure the factoring is correct, you can expand the factored form: \[ 9x^{2}(2x) + 9x^{2}(1) = 18x^{3} + 9x^{2} \] This matches the original expression, confirming that the factoring is accurate. ### Summary By identifying and factoring out the greatest common factor, the expression \( 18x^{3} + 9x^{2} \) simplifies to: \[ 9x^{2}(2x + 1) \]