Problem 92 - Introductory Algebra for College Students

Question

Factor each polynomial. $$8 x^{5}(x+2)-10 x^{3}(x+2)-2 x^{2}(x+2)$$

Step-by-Step Solution

Answer

The factorized form of the given polynomial \(8x^5(x+2) - 10x^3(x+2) - 2x^2(x+2)\) is \(x^2(x+2)[2(4x^3 - 5x - 1)]\).

Step 1: Identifying Common Factors

Begin by looking at all the three terms of the given polynomial, which are \(8x^5(x+2)\), \(-10x^3(x+2)\) and \(-2x^2(x+2)\). It can be noticed that all terms have a common factor of \(x^2\) and \(x+2\). The common factors identified here are \(x^2\) and \(x+2\). Note this down.

Step 2: Factoring Out the Common Factor

Now, factor out these common factors \(x^2\) and \(x+2\) from every term. The resulting expression is: \[x^2(x+2)[8x^3 - 10x - 2]\]

Step 3: Simplify the Resulting Expression

Simplify the expression inside the square brackets. The final step is this simplification to provide the final factored form of the given polynomial.Even within the bracket, we see a common factor of 2: \[x^2(x+2)[2(4x^3 - 5x - 1)]\]This is a neat and final factored form of the given polynomial.