Solve 4x^4-2x^3-6x^2+2x+6div2x^2+x

$$4 x ^ { 4 } - 2 x ^ { 3 } - 6 x ^ { 2 } + 2 x + 6 \div 2 x ^ { 2 } + x$$

Evaluate

$$x\left(x+1\right)\left(4x^{2}-6x+3\right)$$

Solution Steps

Divide 6 by 2 to get 3.

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

Combine -6x^{2} and 3x^{2} to get -3x^{2}.

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

Combine 2x and x to get 3x.

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

Factor

$$x\left(x+1\right)\left(4x^{2}-6x+3\right)$$

Solution Steps

Factor out x.

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

Consider 4x^{3}-2x^{2}-6x+2+3x+1. Multiply and combine like terms.

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

Consider 4x^{3}-2x^{2}-3x+3. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 3 and q divides the leading coefficient 4. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(4x^{2}-6x+3\right)$$

Rewrite the complete factored expression. Polynomial 4x^{2}-6x+3 is not factored since it does not have any rational roots.

$$x\left(x+1\right)\left(4x^{2}-6x+3\right)$$