Solve 7z^5+14z^4-21z^3

$$7 z ^ { 5 } + 14 z ^ { 4 } - 21 z ^ { 3 }$$

Factor

$$7\left(z-1\right)\left(z+3\right)z^{3}$$

Solution Steps

Factor out 7.

$$7\left(z^{5}+2z^{4}-3z^{3}\right)$$

Consider z^{5}+2z^{4}-3z^{3}. Factor out z^{3}.

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

Consider z^{2}+2z-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.

$$a+b=2 ab=1\left(-3\right)=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=3$$

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

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

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

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

Factor out common term z-1 by using distributive property.

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

Rewrite the complete factored expression.

$$7z^{3}\left(z-1\right)\left(z+3\right)$$

Evaluate

$$7\left(z-1\right)\left(z+3\right)z^{3}$$