Problem 36 - Introductory Algebra for College Students

Question

Divide: $$\frac{27 x^{3}-8}{3 x+2}$$

Step-by-Step Solution

Answer

The result of the division is \(9x^2 + 6x + 4\).

Step 1: Rewrite the Cubic Expression

Rewrite the numerator \((27x^3 - 8)\) as \(am^3 - bn^3\), where \(a = 3\), \(m = x\), \(b = 2\), and \(n = 1\). So that it takes the form of the difference of cubes.

Step 2: Apply the difference of cubes formula

Use the difference of cube formula, which is \(am^3 - bn^3 = (am - bn)(a^2m^2 + abmn + b^2n^2)\). This gives us \((3x - 2)(9x^2 + 6x + 4)\) when we substitute \(a = 3\), \(m = x\), \(b = 2\), and \(n = 1\) into this equation.

Step 3: Perform the Division

Divide the resulting expression \((3x^2 + 6x + 4)\) by the original denominator \((3x + 2)\). From the difference of cubes result, we can easily see that the \((3x + 2)\) parts cancel out, leaving only the second part of the expanded equation, which is \(9x^2 + 6x + 4\).