Solve 5y^4+25y^3-70y^2

$$5 y ^ { 4 } + 25 y ^ { 3 } - 70 y ^ { 2 }$$

Factor

$$5\left(y-2\right)\left(y+7\right)y^{2}$$

Solution Steps

Factor out 5.

$$5\left(y^{4}+5y^{3}-14y^{2}\right)$$

Consider y^{4}+5y^{3}-14y^{2}. Factor out y^{2}.

$$y^{2}\left(y^{2}+5y-14\right)$$

Consider y^{2}+5y-14. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-14. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-14\right)=-14$$

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 -14.

$$-1,14 -2,7$$

Calculate the sum for each pair.

$$-1+14=13 -2+7=5$$

The solution is the pair that gives sum 5.

$$a=-2 b=7$$

Rewrite y^{2}+5y-14 as \left(y^{2}-2y\right)+\left(7y-14\right).

$$\left(y^{2}-2y\right)+\left(7y-14\right)$$

Factor out y in the first and 7 in the second group.

$$y\left(y-2\right)+7\left(y-2\right)$$

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

$$\left(y-2\right)\left(y+7\right)$$

Rewrite the complete factored expression.

$$5y^{2}\left(y-2\right)\left(y+7\right)$$

Evaluate

$$5\left(y-2\right)\left(y+7\right)y^{2}$$