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Chapter 6: Problem 29

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. $$9 y^{4}+27 y^{6}$$

Short Answer

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

Step by step solution

01

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 \(9y^{4}\).
02

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, \(9y^{4}+27y^{6} = 9y^{4}(1 + 3y^{2}) \).
03

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.

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Most popular questions from this chapter

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$9 y^{2}-64$$

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