Solve 5y^4+8y^3-4y^2

$$5 y ^ { 4 } + 8 y ^ { 3 } - 4 y ^ { 2 }$$

Factor

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

Solution Steps

Factor out y^{2}.

$$y^{2}\left(5y^{2}+8y-4\right)$$

Consider 5y^{2}+8y-4. Factor the expression by grouping. First, the expression needs to be rewritten as 5y^{2}+ay+by-4. To find a and b, set up a system to be solved.

$$a+b=8 ab=5\left(-4\right)=-20$$

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 -20.

$$-1,20 -2,10 -4,5$$

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 8.

$$a=-2 b=10$$

Rewrite 5y^{2}+8y-4 as \left(5y^{2}-2y\right)+\left(10y-4\right).

$$\left(5y^{2}-2y\right)+\left(10y-4\right)$$

Factor out y in the first and 2 in the second group.

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

Factor out common term 5y-2 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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