Solve 6z^4-24z^3+18z^2

$$6 z ^ { 4 } - 24 z ^ { 3 } + 18 z ^ { 2 }$$

Factor

$$6\left(z-3\right)\left(z-1\right)z^{2}$$

Solution Steps

Factor out 6.

$$6\left(z^{4}-4z^{3}+3z^{2}\right)$$

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

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

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

$$a+b=-4 ab=1\times 3=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-3 b=-1$$

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

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

Factor out z in the first and -1 in the second group.

$$z\left(z-3\right)-\left(z-3\right)$$

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

$$\left(z-3\right)\left(z-1\right)$$

Rewrite the complete factored expression.

$$6z^{2}\left(z-3\right)\left(z-1\right)$$

Evaluate

$$6\left(z-3\right)\left(z-1\right)z^{2}$$