Solve 5y^5-5y^4-10y^3

$$5 y ^ { 5 } - 5 y ^ { 4 } - 10 y ^ { 3 }$$

Factor

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

Solution Steps

Factor out 5.

$$5\left(y^{5}-y^{4}-2y^{3}\right)$$

Consider y^{5}-y^{4}-2y^{3}. Factor out y^{3}.

$$y^{3}\left(y^{2}-y-2\right)$$

Consider y^{2}-y-2. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-2. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-2\right)=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-2 b=1$$

Rewrite y^{2}-y-2 as \left(y^{2}-2y\right)+\left(y-2\right).

$$\left(y^{2}-2y\right)+\left(y-2\right)$$

Factor out y in y^{2}-2y.

$$y\left(y-2\right)+y-2$$

Factor out common term y-2 by using distributive property.

$$\left(y-2\right)\left(y+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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