Q:

polynomial expressions with matching​

Accepted Solution

A:
Answer:The difference of 27x³ and 8y³ → [3x - 2y][9x² + 6xy + 4y²]The difference of 27x³ and 64y³ → [3x - 4y][9x² + 12xy + 16y²]The sum of 27x³ and 64y³ → [3x + 4y][9x² - 12xy + 16y²]The sum of 27x³ and 8y³ → [3x + 2y][9x² - 6xy + 4y²]Step-by-step explanation:For the first two, we use the Difference of Cubes [(a³ - b³)(a² + 2ab + b²)] first by taking the cube root of the given expression to get our first factor[s]:[tex]\displaystyle 3x - 2y = \sqrt[3]{27x^3 - 8y^3} \\ 3x - 4y = \sqrt[3]{27x^3 - 64y^3}[/tex]Then, use the acronym of SOAP {whether the next operations symbols will be negative or positive [SAME (your cube-rooted factor has an IDENTICAL OPERATION SYMBOL as your given expression), OPPOSITE (the first sign in your second factor in the second set of parentheses will be the opposite of what the sign in your given expression, which will be a plus sign), ALWAYS POSITIVE (the last sign in your second factor in the second set of parentheses will ALWAYS stay positive NO MATTER WHAT)]} to get the second factor:Given: 27x³ - 64y³[3x - 4y][9x² + 12xy + 16y²] ↑ ↑ ↑same as opposite ALWAYS POSITIVE given of givenGiven: 27x³ - 8y³[3x - 2y][9x² + 6xy + 4y²] ↑ ↑ ↘same as opposite ALWAYS POSITIVE given of givenNow, for the last two, we use the Sum of Cubes [(a³ + b³)(a² - 2ab + b²)] first by taking the cube root of the given expression to get our first factor[s]:[tex]\displaystyle 3x + 2y = \sqrt[3]{27x^3 + 8y^3} \\ 3x + 4y = \sqrt[3]{27x^3 + 64y^3}[/tex]Then, use the acronym of SOAP {whether the next operations symbols will be negative or positive [SAME (your cube-rooted factor has an IDENTICAL OPERATION SYMBOL as your given expression), OPPOSITE (the first sign in your second factor in the second set of parentheses will be the opposite of what the sign is in your given expression, which will be a minus sign), ALWAYS POSITIVE (the last sign in your second factor in the second set of parentheses will ALWAYS stay positive NO MATTER WHAT)]} to get the second factor:Given: 27x³ + 64y³[3x + 4y][9x² - 12xy + 16y²] ↑ ↑ ↘same as opposite ALWAYS POSITIVE given of givenGiven: 27x³ + 8y³[3x + 2y][9x² - 6xy + 4y²] ↑ ↑ ↘same as opposite ALWAYS POSITIVE given of givenI am delighted to assist you anytime!* As you can see, when using the Difference\Sum of Cubes, SOAP can vary depending on how an expression is given to you. They resemble each other though.