Solve 1/6p^3+1/2p^2+1/3p

$$\frac { 1 } { 6 } p ^ { 3 } + \frac { 1 } { 2 } p ^ { 2 } + \frac { 1 } { 3 } p$$

Factor

$$\frac{p\left(p+1\right)\left(p+2\right)}{6}$$

Solution Steps

Factor out \frac{1}{6}.

$$\frac{p^{3}+3p^{2}+2p}{6}$$

Consider p^{3}+3p^{2}+2p. Factor out p.

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

Consider p^{2}+3p+2. Factor the expression by grouping. First, the expression needs to be rewritten as p^{2}+ap+bp+2. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

Rewrite p^{2}+3p+2 as \left(p^{2}+p\right)+\left(2p+2\right).

$$\left(p^{2}+p\right)+\left(2p+2\right)$$

Factor out p in the first and 2 in the second group.

$$p\left(p+1\right)+2\left(p+1\right)$$

Factor out common term p+1 by using distributive property.

$$\left(p+1\right)\left(p+2\right)$$

Rewrite the complete factored expression.

$$\frac{p\left(p+1\right)\left(p+2\right)}{6}$$

Evaluate

$$\frac{p\left(p+1\right)\left(p+2\right)}{6}$$