Solve 12z^4-9z^3-30z^2

$$12 z ^ { 4 } - 9 z ^ { 3 } - 30 z ^ { 2 }$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(4z^{4}-3z^{3}-10z^{2}\right)$$

Consider 4z^{4}-3z^{3}-10z^{2}. Factor out z^{2}.

$$z^{2}\left(4z^{2}-3z-10\right)$$

Consider 4z^{2}-3z-10. Factor the expression by grouping. First, the expression needs to be rewritten as 4z^{2}+az+bz-10. To find a and b, set up a system to be solved.

$$a+b=-3 ab=4\left(-10\right)=-40$$

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 -40.

$$1,-40 2,-20 4,-10 5,-8$$

Calculate the sum for each pair.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

The solution is the pair that gives sum -3.

$$a=-8 b=5$$

Rewrite 4z^{2}-3z-10 as \left(4z^{2}-8z\right)+\left(5z-10\right).

$$\left(4z^{2}-8z\right)+\left(5z-10\right)$$

Factor out 4z in the first and 5 in the second group.

$$4z\left(z-2\right)+5\left(z-2\right)$$

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

$$\left(z-2\right)\left(4z+5\right)$$

Rewrite the complete factored expression.

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

Evaluate

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