Solve f(x)=x^4+3x^3+x^2/3+x

$$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } + \frac { x ^ { 2 } } { 3 } + x$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{4}+3x^{3}+x times \frac{3}{3}.

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

Since \frac{3\left(x^{4}+3x^{3}+x\right)}{3} and \frac{x^{2}}{3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(x^{4}+3x^{3}+x\right)+x^{2}.

$$\frac{3x^{4}+9x^{3}+3x+x^{2}}{3}$$

Factor

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

Solution Steps

Factor out \frac{1}{3}.

$$\frac{3x^{4}+9x^{3}+x^{2}+3x}{3}$$

Consider 3x^{4}+9x^{3}+x^{2}+3x. Factor out x.

$$x\left(3x^{3}+9x^{2}+x+3\right)$$

Consider 3x^{3}+9x^{2}+x+3. Do the grouping 3x^{3}+9x^{2}+x+3=\left(3x^{3}+9x^{2}\right)+\left(x+3\right), and factor out 3x^{2} in 3x^{3}+9x^{2}.

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

Factor out common term x+3 by using distributive property.

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

Rewrite the complete factored expression. Polynomial 3x^{2}+1 is not factored since it does not have any rational roots.

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