Problem 121 - Introductory Algebra for College Students

Question

Factor completely. $$y^{7}+y$$

Step-by-Step Solution

Answer

The complete factorization of \(y^{7}+y\) is \(y(y^{6}+1)\).

Step 1 - Identifying the Greatest Common Factor (GCF)

Here, the GCF is a common term that can be factored out from every term of the given polynomial. For the expression \(y^{7}+y\), the GCF is \(y\), because it is a factor of both \(y^{7}\) and \(y\).

Step 2 - Factoring Out the GCF

Divide each term of the polynomial by the GCF and rewrite the polynomial as the product of the GCF and the resulting expression. So, \(y^{7}+y\) will be factored as \(y(y^{6}+1)\).

Step 3 - Checking for Further Factorization

Now, check if the expression \(y^{6}+1\) can be factored further. The expression \(a^{2}+1\) can't be factored, but \(a^{2}+1 = (a+ i)(a- i)\) in complex numbers where 'i' is a imaginary unit with the property that \(i^{2} = -1\). So, \(y^{6}+1\) can be re-written as \((y^{2}+1)(y^{4}-y^{2}+1)\). But considering real numbers, \(y^{6}+1\) can't be factored further.