Solve 3z^6-30z^5+63z^4

$$3 z ^ { 6 } - 30 z ^ { 5 } + 63 z ^ { 4 }$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(z^{6}-10z^{5}+21z^{4}\right)$$

Consider z^{6}-10z^{5}+21z^{4}. Factor out z^{4}.

$$z^{4}\left(z^{2}-10z+21\right)$$

Consider z^{2}-10z+21. Factor the expression by grouping. First, the expression needs to be rewritten as z^{2}+az+bz+21. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 21=21$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-7 b=-3$$

Rewrite z^{2}-10z+21 as \left(z^{2}-7z\right)+\left(-3z+21\right).

$$\left(z^{2}-7z\right)+\left(-3z+21\right)$$

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

$$z\left(z-7\right)-3\left(z-7\right)$$

Factor out common term z-7 by using distributive property.

$$\left(z-7\right)\left(z-3\right)$$

Rewrite the complete factored expression.

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

Evaluate

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