Factor y^5-y^3-y^2+1

$y^{5} - y^{3} - y^{2} + 1$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(y^{5} - y^{3}) - y^{2} + 1$

Factor out the greatest common factor (GCF) from each group.

$y^{3} (y^{2} - 1) - (y^{2} - 1)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $y^{2} - 1$ .

$(y^{2} - 1) (y^{3} - 1)$

Step 3

Rewrite $1$ as $1^{2}$ .

$(y^{2} - 1^{2}) (y^{3} - 1)$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$(y + 1) (y - 1) (y^{3} - 1)$

Step 5

Rewrite $1$ as $1^{3}$ .

$(y + 1) (y - 1) (y^{3} - 1^{3})$

Step 6

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 1$ .

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

Step 7

Factor.

Simplify.

Multiply $y$ by $1$ .

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

One to any power is one.

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

Remove unnecessary parentheses.

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

Step 8

Combine exponents.

Raise $y - 1$ to the power of $1$ .

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

Raise $y - 1$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Factor y^5-y^3-y^2+1

Related Questions

Support LCMGCF.com by disabling your adblocker.