Solve c^5-12c^4+35c^3

$$c ^ { 5 } - 12 c ^ { 4 } + 35 c ^ { 3 }$$

Factor

$$\left(c-7\right)\left(c-5\right)c^{3}$$

Solution Steps

Factor out c^{3}.

$$c^{3}\left(c^{2}-12c+35\right)$$

Consider c^{2}-12c+35. Factor the expression by grouping. First, the expression needs to be rewritten as c^{2}+ac+bc+35. To find a and b, set up a system to be solved.

$$a+b=-12 ab=1\times 35=35$$

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 35.

$$-1,-35 -5,-7$$

Calculate the sum for each pair.

$$-1-35=-36 -5-7=-12$$

The solution is the pair that gives sum -12.

$$a=-7 b=-5$$

Rewrite c^{2}-12c+35 as \left(c^{2}-7c\right)+\left(-5c+35\right).

$$\left(c^{2}-7c\right)+\left(-5c+35\right)$$

Factor out c in the first and -5 in the second group.

$$c\left(c-7\right)-5\left(c-7\right)$$

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

$$\left(c-7\right)\left(c-5\right)$$

Rewrite the complete factored expression.

$$c^{3}\left(c-7\right)\left(c-5\right)$$

Evaluate

$$\left(c-7\right)\left(c-5\right)c^{3}$$