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Chapter 6: Q 268 (page 613)

In the following exercises, factor completely.

y6+1

Short Answer

Expert verified

The expression is factored asy6+1=(y2+1)(y4-y2+1)

Step by step solution

01

Step 1. Given Information

Consider the expression y6+1

The objective is to factor the expression completely.

02

Step 2. Factor using the sum of cubes

The sum of cubes can be factored as a3+b3=(a+b)(a2-ab+b2)

In the given binomial y6+1, the first term y6is the cube of y2and the second term 1is the cube of 1. So it can be factored as

role="math" localid="1644654698408" y6+1=(y2)3+(1)3=(y2+1){(y2)2-y2路1+(1)2}=(y2+1)(y4-y2+1)

03

Step 3. Check the factors

Multiply the factors to check the solution

(y2+1)(y4-y2+1)=y2(y4-y2+1)+1(y4-y2+1)=y6-y4+y2+y4-y2+1=y6+1

And we get the given expression. So the expression is correctly factored.

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