Solve 8x^5+2x^4-6x^3

$$8 x ^ { 5 } + 2 x ^ { 4 } - 6 x ^ { 3 }$$

Factor

$$2\left(4x-3\right)\left(x+1\right)x^{3}$$

Solution Steps

Factor out 2.

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

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

$$x^{3}\left(4x^{2}+x-3\right)$$

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

$$a+b=1 ab=4\left(-3\right)=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 1.

$$a=-3 b=4$$

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

$$\left(4x^{2}-3x\right)+\left(4x-3\right)$$

Factor out x in 4x^{2}-3x.

$$x\left(4x-3\right)+4x-3$$

Factor out common term 4x-3 by using distributive property.

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

Rewrite the complete factored expression.

$$2x^{3}\left(4x-3\right)\left(x+1\right)$$

Evaluate

$$2\left(4x-3\right)\left(x+1\right)x^{3}$$