Problem 58 - Algebra and Trigonometry

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3}+64$$

Step-by-Step Solution

Answer

The factorised form of the expression \(x^3 + 64\) using the formula for the sum of cubes is \((x + 4)(x^2 - 4x + 16)\).

Step 1: Identify A and B

From the expression \(x^3 + 64\), we see that the cube root of \(x^3\) is \(x\) and the cube root of \(64\) is \(4\). Therefore, \(A = x\) and \(B = 4\).

Step 2: Apply the formula for the sum of cubes

Substitute \(A = x\) and \(B = 4\) into the formula \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\). This results in \((x + 4)(x^2 - 4x + 16)\).

Step 3: Check the solution

If the solution is correct, then multiplying the factored form \((x + 4)(x^2 - 4x + 16)\) should return the original expression \(x^3 + 64\). Perform this multiplication to verify your solution.