Solve 2x^3+4x^2+1/2x^3-3/2

$$2 x ^ { 3 } + 4 x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } - \frac { 3 } { 2 }$$

Factor

$$\frac{\left(x+1\right)\left(5x^{2}+3x-3\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{4x^{3}+8x^{2}+x^{3}-3}{2}$$

Consider 4x^{3}+8x^{2}+x^{3}-3. Multiply and combine like terms.

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

Consider 5x^{3}+8x^{2}-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 5. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(5x^{2}+3x-3\right)$$

Rewrite the complete factored expression. Polynomial 5x^{2}+3x-3 is not factored since it does not have any rational roots.

$$\frac{\left(x+1\right)\left(5x^{2}+3x-3\right)}{2}$$

Evaluate

$$\frac{5x^{3}}{2}+4x^{2}-\frac{3}{2}$$

Solution Steps

Combine 2x^{3} and \frac{1}{2}x^{3} to get \frac{5}{2}x^{3}.

$$\frac{5}{2}x^{3}+4x^{2}-\frac{3}{2}$$