Problem 29 - 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. $$9 y^{4}+27 y^{6}$$

Step-by-Step Solution

Answer

The factorized form of the polynomial is \(9y^{4}(1 + 3y^{2})\).

Step 1: Identify the Greatest Common Factor

First, examine both terms to find the greatest common factor. Here the numerical coefficients are 9 and 27. The greatest common factor (GCF) of 9 and 27 is 9. Also, both terms are multiples of \(y\), and the smallest power of \(y\) present is \(y^{4}\). Therefore, the GCF of these two terms is \(9y^{4}\).

Step 2: Divide each term by the GCF

Next, divide each term of the polynomial by the GCF. This gives \(9y^{4}/9y^{4} = 1\) and \(27y^{6}/9y^{4} = 3y^{2}\). Therefore, when factored out, \(9y^{4}+27y^{6} = 9y^{4}(1 + 3y^{2}) \).

Step 3: Check the factoring

Finally, the validity of the factoring can be checked by distributing \(9y^{4}\) back into the brackets. Doing this operation, \(9y^{4}*1 + 9y^{4}*3y^{2}\), results in the original polynomial \(9y^{4}+27y^{6}\). This means the factoring was carried out correctly.