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

Ask by Hodgson Washington. in the United States
Jan 21,2025

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Answer

The expression \( 18x^{3} + 9x^{2} \) factors to \( 9x^{2}(2x + 1) \).

Solution

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: \[ 9x^{2}(2x + 1) \]

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Bonus Knowledge

To factor the expression \( 18x^{3} + 9x^{2} \), start by identifying the greatest common factor (GCF) of both terms, which is \( 9x^{2} \). Factoring this out gives you \( 9x^{2}(2x + 1) \). This can help make further calculations more manageable, especially if you are solving equations or looking for roots. One common mistake is to skip finding the GCF and attempt to factor blindly. Always check for common factors first; it simplifies your expression and reduces the chances of errors later on. Remember, the easier the starting point, the smoother the journey to solving the problem!

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