Question

question
7
place the correct answer in each box.
consider the sum of cubes identity:
(a^3 + b^3 = (a + b)(a^2 - ab + b^2)).
for the polynomial (8x^3 + 27),
(a = square) and (b = square).

Explanation:

Step1: Identify \(a^3\) and \(b^3\)

The polynomial is \(8x^3 + 27\). We can rewrite it as \((2x)^3 + 3^3\). So, comparing with \(a^3 + b^3\), we need to find \(a\) and \(b\) such that \(a^3=(2x)^3\) and \(b^3 = 3^3\).

Step2: Solve for \(a\) and \(b\)

For \(a^3=(2x)^3\), taking the cube - root of both sides, we get \(a = 2x\) (since the cube - root of \(y^3\) is \(y\)). For \(b^3=3^3\), taking the cube - root of both sides, we get \(b = 3\) (since the cube - root of \(z^3\) is \(z\)).

Answer:

\(a = 2x\) and \(b = 3\)