Solve 3x^6-3x^5+12x^5-12x^4

$$3 x ^ { 6 } - 3 x ^ { 5 } + 12 x ^ { 5 } - 12 x ^ { 4 }$$

Factor

$$3\left(x-1\right)\left(x+4\right)x^{4}$$

Solution Steps

Factor out 3.

$$3\left(x^{6}-x^{5}+4x^{5}-4x^{4}\right)$$

Consider x^{6}-x^{5}+4x^{5}-4x^{4}. Factor out x^{4}.

$$x^{4}\left(x^{2}-x+4x-4\right)$$

Consider x^{2}-x+4x-4. Multiply and combine like terms.

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

Consider x^{2}+3x-4. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\left(-4\right)=-4$$

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. List all such integer pairs that give product -4.

$$-1,4 -2,2$$

Calculate the sum for each pair.

$$-1+4=3 -2+2=0$$

The solution is the pair that gives sum 3.

$$a=-1 b=4$$

Rewrite x^{2}+3x-4 as \left(x^{2}-x\right)+\left(4x-4\right).

$$\left(x^{2}-x\right)+\left(4x-4\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-1\right)+4\left(x-1\right)$$

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

$$\left(x-1\right)\left(x+4\right)$$

Rewrite the complete factored expression.

$$3x^{4}\left(x-1\right)\left(x+4\right)$$

Evaluate

$$3\left(x-1\right)\left(x+4\right)x^{4}$$

Solution Steps

Combine -3x^{5} and 12x^{5} to get 9x^{5}.

$$3x^{6}+9x^{5}-12x^{4}$$