Solve 4z^4-14z^3-30z^2

$$4 z ^ { 4 } - 14 z ^ { 3 } - 30 z ^ { 2 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(2z^{4}-7z^{3}-15z^{2}\right)$$

Consider 2z^{4}-7z^{3}-15z^{2}. Factor out z^{2}.

$$z^{2}\left(2z^{2}-7z-15\right)$$

Consider 2z^{2}-7z-15. Factor the expression by grouping. First, the expression needs to be rewritten as 2z^{2}+az+bz-15. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\left(-15\right)=-30$$

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. List all such integer pairs that give product -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -7.

$$a=-10 b=3$$

Rewrite 2z^{2}-7z-15 as \left(2z^{2}-10z\right)+\left(3z-15\right).

$$\left(2z^{2}-10z\right)+\left(3z-15\right)$$

Factor out 2z in the first and 3 in the second group.

$$2z\left(z-5\right)+3\left(z-5\right)$$

Factor out common term z-5 by using distributive property.

$$\left(z-5\right)\left(2z+3\right)$$

Rewrite the complete factored expression.

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

Evaluate

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