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 \(b=y\). We need to substitute \(a\) and \(b\) into the formula \((a + b)(a^2 - ab + b^2)\) to find the correct product.

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

First, calculate \(a^2\):
\(a^2=(2x)^2 = 4x^2\)
Then, calculate \(ab\):
\(ab=(2x)(y)=2xy\)
And \(b^2=y^2\)
Now, substitute these into \(a^2 - ab + b^2\):
\(a^2 - ab + b^2=4x^2-2xy + y^2\)
And the first factor is \(a + b=(2x)+y = 2x + y\)
So the product should be \((2x + y)(4x^2-2xy + y^2)\)

Answer:

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