Problem 62 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$8 u^{2}-16 u v^{2}+3 u v-6 v^{3}$$

Step-by-Step Solution

Answer

The short answer is: \( (8u + 3v)(u - 2v^2) \)

Step 1: Identify the groups

First, separate the given expression into two groups: \( (8u^2 - 16u v^2) + (3uv - 6v^3) \)

Step 2: Factor out the GCD from each group

Factor out the GCD from each group: Group 1: \(8u^2 - 16u v^2\) has a GCD of \(8u\). After factoring, we get: \( 8u(u - 2v^2) \) Group 2: \(3uv - 6v^3\) has a GCD of \(3v\). After factoring, we get: \( 3v(u - 2v^2) \) Now write the factored expression after taking out GCD from both groups: \( 8u(u - 2v^2) + 3v(u - 2v^2) \)

Step 3: Factor by grouping

Since both terms have a common factor of \((u - 2v^2)\), we can factor that out: \( (8u + 3v)(u - 2v^2) \) So the factored expression by grouping is: \( (8u + 3v)(u - 2v^2) \)