Question

tomas learned that the product of the polynomials ((a + b)(a^2 - ab + b^2)) was a special pattern that would result in a sum of cubes, (a^3 + b^3). his teacher put four products on the board and asked the class to identify which product would result in a sum of cubes if (a = 2x) and (b = y). which product should tomas choose? (\bigcirc) ((2x + y)(2x^2 + 2xy - y^2)) (\bigcirc) ((2x + y)(4x^2 + 2xy - y^2)) (\bigcirc) ((2x + y)(4x^2 - 2xy + y^2)) (\bigcirc) ((2x + y)(2x^2 - 2xy + y^2))

Explanation:

Step1: Recall the sum of cubes formula

The formula for the sum of cubes is \((a + b)(a^2 - ab + b^2)=a^3 + b^3\). We are given that \(a = 2x\) and \(b=y\). So we need to substitute \(a = 2x\) and \(b = y\) into the formula \((a + b)(a^2 - ab + b^2)\).

Step2: Substitute \(a = 2x\) and \(b=y\) into \(a^2 - ab + b^2\)

First, calculate \(a^2\): If \(a = 2x\), then \(a^2=(2x)^2 = 4x^2\).
Next, calculate \(ab\): If \(a = 2x\) and \(b = y\), then \(ab=(2x)\times y=2xy\).
Then, \(b^2=y^2\).
Substitute these into \(a^2 - ab + b^2\), we get \(4x^2-2xy + y^2\).
And the first factor is \(a + b=2x + y\). So the product should be \((2x + y)(4x^2-2xy + y^2)\).

Answer:

\((2x + y)(4x^2-2xy + y^2)\) (the third option)