Problem 38 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$26 y^{5}-13 y^{3}+39 y^{2}$$

Step-by-Step Solution

Answer

The factored form of \(26y^5 - 13y^3 + 39y^2\) is \(13y^2(2y^3 - y + 3)\).

Step 1: Find the Greatest Common Factor

Look for the greatest common factor (GCF) in the coefficients (numbers in front of the variable) and in the variable part. In this case, the GCF of the coefficients 26, -13, 39 is 13, and for the variables \(y^5\), \(y^3\), \(y^2\) is \(y^2\). Therefore, the GCF is \(13y^2\).

Step 2: Factor out GCF

The next step is to take out the GCF from each term. It will have the same value for every term. So, \(26y^5 - 13y^3 + 39y^2\) becomes \(13y^2(2y^3 - y + 3)\).

Step 3: Check the Factored Equation

To ensure that the factoring is correct, one can distribute the factor back into the parentheses. If it returns the initial polynomial, the factoring is correct.