Solve x^4-12x^3+35x^2

$$x ^ { 4 } - 12 x ^ { 3 } + 35 x ^ { 2 }$$

Factor

$$\left(x-7\right)\left(x-5\right)x^{2}$$

Solution Steps

Factor out x^{2}.

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

Consider x^{2}-12x+35. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+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 x^{2}-12x+35 as \left(x^{2}-7x\right)+\left(-5x+35\right).

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

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

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

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

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

Rewrite the complete factored expression.

$$x^{2}\left(x-7\right)\left(x-5\right)$$

Evaluate

$$\left(x-7\right)\left(x-5\right)x^{2}$$