# [[SOAP]] (Mathematics > Algebra) ## Definition **SOAP** is a mnemonic used to help remember the factoring patterns of special polynomials, specifically for factoring the sum or difference of cubes. SOAP stands for **Same, Opposite, Always Positive**, which refers to the signs in the factored form of these expressions. The general forms of the sum and difference of cubes are: - **Sum of cubes**: $a^3 + b^3$ $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $ - **Difference of cubes**: $a^3 - b^3$ $ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $ ## Key Concepts - **Same**: The first sign in the factored form is the **same** as in the original expression. - **Opposite**: The second sign is the **opposite** of the original expression's sign. - **Always Positive**: The third term in the second factor is **always positive**, regardless of the original expression. ## Important Properties 1. **Sum of cubes**: For $a^3 + b^3$, the factored form is $(a + b)(a^2 - ab + b^2)$. The signs follow SOAP: - **Same**: The first sign in the factor is the same as the original ($+$). - **Opposite**: The second sign in the quadratic factor is opposite ($-$). - **Always Positive**: The third term is always positive ($+$). 2. **Difference of cubes**: For $a^3 - b^3$, the factored form is $(a - b)(a^2 + ab + b^2)$. Again, following SOAP: - **Same**: The first sign is the same as the original ($-$). - **Opposite**: The second sign is opposite ($+$). - **Always Positive**: The third term is always positive ($+$). ## Essential Formulas - **Sum of cubes**: $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $ - **Difference of cubes**: $ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $ ## Core Examples 1. **Factoring a sum of cubes**: Factor $x^3 + 27$. \[ x^3 + 27 = x^3 + 3^3 = (x + 3)(x^2 - 3x + 9) \] Using SOAP: Same ($+$), Opposite ($-$), Always Positive ($+$). 2. **Factoring a difference of cubes**: Factor $8x^3 - 125$. \[ 8x^3 - 125 = (2x)^3 - 5^3 = (2x - 5)((2x)^2 + (2x)(5) + 5^2) \] Simplifying: \[ (2x - 5)(4x^2 + 10x + 25) \] Using SOAP: Same ($-$), Opposite ($+$), Always Positive ($+$). ## Related Theorems/Rules - **Difference of Squares**: Another factoring pattern for $a^2 - b^2$, though unrelated to cubes: $ a^2 - b^2 = (a - b)(a + b) $ ## Common Pitfalls - **Incorrect signs**: It's easy to get confused with the signs in the quadratic factor. Use the SOAP mnemonic to ensure correct sign placement. - **Applying SOAP to incorrect forms**: Remember that SOAP applies only to **sum** or **difference** of cubes, not to squares or other powers. ## Related Topics - [[Factoring Polynomials]] - [[Difference of Squares]] - [[Quadratic Expressions]] ## Quick Review Questions 1. Factor $x^3 - 64$ using the SOAP method. 2. What are the factored forms of $a^3 + b^3$ and $a^3 - b^3$? Explain how SOAP helps in factoring. 3. Why is the third term in the quadratic factor always positive in the sum or difference of cubes? This explanation uses the mnemonic SOAP to clearly explain the process for factoring sums and differences of cubes, making it easier to remember the correct signs in the factorization.