Question

1. $y = 27x^4 + 8x$

Explanation:

Step1: Identify common factor

Both terms \(27x^4\) and \(8x\) have a common factor of \(x\).

Step2: Factor out the common factor

Factor out \(x\) from the expression: \(y = x(27x^3 + 8)\).

Step3: Recognize sum of cubes

Notice that \(27x^3=(3x)^3\) and \(8 = 2^3\), so \(27x^3 + 8\) is a sum of cubes, which factors as \((a + b)(a^2 - ab + b^2)\) where \(a = 3x\) and \(b = 2\).

Step4: Factor the sum of cubes

Applying the sum of cubes formula: \(27x^3 + 8=(3x + 2)(9x^2 - 6x + 4)\).

Step5: Combine the factors

Substitute back into the expression for \(y\): \(y = x(3x + 2)(9x^2 - 6x + 4)\).

Answer:

\(y = x(3x + 2)(9x^2 - 6x + 4)\)