Solve s^5-5s^4+6s^3

$$s ^ { 5 } - 5 s ^ { 4 } + 6 s ^ { 3 }$$

Factor

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

Solution Steps

Factor out s^{3}.

$$s^{3}\left(s^{2}-5s+6\right)$$

Consider s^{2}-5s+6. Factor the expression by grouping. First, the expression needs to be rewritten as s^{2}+as+bs+6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 6=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -5.

$$a=-3 b=-2$$

Rewrite s^{2}-5s+6 as \left(s^{2}-3s\right)+\left(-2s+6\right).

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

Factor out s in the first and -2 in the second group.

$$s\left(s-3\right)-2\left(s-3\right)$$

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

$$\left(s-3\right)\left(s-2\right)$$

Rewrite the complete factored expression.

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

Evaluate

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