Solve -18y^3+5y^4+9y^2

$$- 18 y ^ { 3 } + 5 y ^ { 4 } + 9 y ^ { 2 }$$

Factor

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

Solution Steps

Factor out y^{2}.

$$y^{2}\left(-18y+5y^{2}+9\right)$$

Consider -18y+5y^{2}+9. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$5y^{2}-18y+9$$

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

$$a+b=-18 ab=5\times 9=45$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 45.

$$-1,-45 -3,-15 -5,-9$$

Calculate the sum for each pair.

$$-1-45=-46 -3-15=-18 -5-9=-14$$

The solution is the pair that gives sum -18.

$$a=-15 b=-3$$

Rewrite 5y^{2}-18y+9 as \left(5y^{2}-15y\right)+\left(-3y+9\right).

$$\left(5y^{2}-15y\right)+\left(-3y+9\right)$$

Factor out 5y in the first and -3 in the second group.

$$5y\left(y-3\right)-3\left(y-3\right)$$

Factor out common term y-3 by using distributive property.

$$\left(y-3\right)\left(5y-3\right)$$

Rewrite the complete factored expression.

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

Evaluate

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