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Chapter 6: Problem 38

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. $$26 y^{5}-13 y^{3}+39 y^{2}$$

Short Answer

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

Step by step solution

01

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 \(13y^2\).
02

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 \(13y^2(2y^3 - y + 3)\).
03

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.

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Most popular questions from this chapter

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$18 x^{3} y+57 x^{2} y^{2}+30 x y^{3}$$

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. $$100 y^{2}-49$$

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. When a factorization requires two factoring techniques, I'm less likely to make errors if I show one technique at a time rather than combining the two factorizations into one step.

Factor completely. $$(y+1)^{3}+1$$

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. $$9 x^{4}+18 x^{3}+6 x^{2}$$
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