Solve 8z^4-34z^3-30z^2

$$8 z ^ { 4 } - 34 z ^ { 3 } - 30 z ^ { 2 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(4z^{4}-17z^{3}-15z^{2}\right)$$

Consider 4z^{4}-17z^{3}-15z^{2}. Factor out z^{2}.

$$z^{2}\left(4z^{2}-17z-15\right)$$

Consider 4z^{2}-17z-15. Factor the expression by grouping. First, the expression needs to be rewritten as 4z^{2}+az+bz-15. To find a and b, set up a system to be solved.

$$a+b=-17 ab=4\left(-15\right)=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -17.

$$a=-20 b=3$$

Rewrite 4z^{2}-17z-15 as \left(4z^{2}-20z\right)+\left(3z-15\right).

$$\left(4z^{2}-20z\right)+\left(3z-15\right)$$

Factor out 4z in the first and 3 in the second group.

$$4z\left(z-5\right)+3\left(z-5\right)$$

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

$$\left(z-5\right)\left(4z+3\right)$$

Rewrite the complete factored expression.

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

Evaluate

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