Solve 4y^5+18y^4-10y^3

$$4 y ^ { 5 } + 18 y ^ { 4 } - 10 y ^ { 3 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(2y^{5}+9y^{4}-5y^{3}\right)$$

Consider 2y^{5}+9y^{4}-5y^{3}. Factor out y^{3}.

$$y^{3}\left(2y^{2}+9y-5\right)$$

Consider 2y^{2}+9y-5. Factor the expression by grouping. First, the expression needs to be rewritten as 2y^{2}+ay+by-5. To find a and b, set up a system to be solved.

$$a+b=9 ab=2\left(-5\right)=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 9.

$$a=-1 b=10$$

Rewrite 2y^{2}+9y-5 as \left(2y^{2}-y\right)+\left(10y-5\right).

$$\left(2y^{2}-y\right)+\left(10y-5\right)$$

Factor out y in the first and 5 in the second group.

$$y\left(2y-1\right)+5\left(2y-1\right)$$

Factor out common term 2y-1 by using distributive property.

$$\left(2y-1\right)\left(y+5\right)$$

Rewrite the complete factored expression.

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

Evaluate

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