Step 1 We have to find the factor of y7+y. Factorization is the process of taking common from the given expressions, the terms which get common we can call it as factor of the expression. Here, y7+y taking y common from y7(y6⋅y) and y y7+y=y6⋅y+y =y(y6+1) Further we can rewrite: y(y6+1)=y((y2)3+13) Step 2 Now applying the formula a3+b3=(a+b)(a2−ab+b2) for the expression ((y2)2+13) Here a=y2,b=1 y7+y=y(y6+1) =y((y2)3+13) =y(y2+1)(y2−y⋅1+1) =y(y2+1)(y2−y+1) Hence, factorization of y7+y is y(y2+1)(y2−y+1) and their factors are the following: y y2+1 y2−y+1

"Factor : y7+y" was answered by our experts

High School
Algebra
Algebra II
Polynomial factorization

aramutselv

Answered question

2021-12-11

Factor :

y^{7} + y

Answer & Explanation

Lynne Trussell

Beginner2021-12-12Added 32 answers

Step 1
We have to find the factor of

y^{7} + y

.
Factorization is the process of taking common from the given expressions, the terms which get common we can call it as factor of the expression.
Here,

y^{7} + y

taking y common from

y^{7} (y^{6} \cdot y)

and y

y^{7} + y = y^{6} \cdot y + y

= y (y^{6} + 1)

Further we can rewrite:

y (y^{6} + 1) = y ({(y^{2})}^{3} + 1^{3})

Step 2
Now applying the formula

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

for the expression

({(y^{2})}^{2} + 1^{3})

Here

a = y^{2}, b = 1

y^{7} + y = y (y^{6} + 1)

= y ({(y^{2})}^{3} + 1^{3})

= y (y^{2} + 1) (y^{2} - y \cdot 1 + 1)

= y (y^{2} + 1) (y^{2} - y + 1)

Hence, factorization of

y^{7} + y

y (y^{2} + 1) (y^{2} - y + 1)

and their factors are the following:
y

y^{2} + 1

y^{2} - y + 1

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