Factor completely using the formula for sum \( x^{3}+1= \) Question Help: D Video

To factor the expression \( x^{3} + 1 \) completely, you can use the **sum of cubes** formula. Here's a step-by-step guide to help you understand the process. ### Sum of Cubes Formula The sum of two cubes can be factored using the following formula: \[ a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}) \] ### Applying the Formula Let's apply this formula to the expression \( x^{3} + 1 \). 1. **Identify \( a \) and \( b \):** Compare \( x^{3} + 1 \) with \( a^{3} + b^{3} \): - \( a = x \) - \( b = 1 \) (since \( 1^{3} = 1 \)) 2. **Plug \( a \) and \( b \) into the formula:** \[ x^{3} + 1 = (x)^{3} + (1)^{3} = (x + 1)(x^{2} - x \cdot 1 + 1^{2}) \] 3. **Simplify the expression:** \[ x^{3} + 1 = (x + 1)(x^{2} - x + 1) \] ### Final Factored Form \[ \boxed{(x + 1)(x^{2} - x + 1)} \] ### Verification (Optional) To ensure that the factoring is correct, you can expand the factored form: \[ (x + 1)(x^{2} - x + 1) = x(x^{2} - x + 1) + 1(x^{2} - x + 1) = x^{3} - x^{2} + x + x^{2} - x + 1 = x^{3} + 1 \] The expansion returns the original expression, confirming that the factoring is correct. --- **Summary:** The expression \( x^{3} + 1 \) can be completely factored using the sum of cubes formula as: \[ x^{3} + 1 = (x + 1)(x^{2} - x + 1) \]