Solve r^4+9r^3+14r^2

$$r ^ { 4 } + 9 r ^ { 3 } + 14 r ^ { 2 }$$

Factor

$$\left(r+2\right)\left(r+7\right)r^{2}$$

Solution Steps

Factor out r^{2}.

$$r^{2}\left(r^{2}+9r+14\right)$$

Consider r^{2}+9r+14. Factor the expression by grouping. First, the expression needs to be rewritten as r^{2}+ar+br+14. To find a and b, set up a system to be solved.

$$a+b=9 ab=1\times 14=14$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 14.

$$1,14 2,7$$

Calculate the sum for each pair.

$$1+14=15 2+7=9$$

The solution is the pair that gives sum 9.

$$a=2 b=7$$

Rewrite r^{2}+9r+14 as \left(r^{2}+2r\right)+\left(7r+14\right).

$$\left(r^{2}+2r\right)+\left(7r+14\right)$$

Factor out r in the first and 7 in the second group.

$$r\left(r+2\right)+7\left(r+2\right)$$

Factor out common term r+2 by using distributive property.

$$\left(r+2\right)\left(r+7\right)$$

Rewrite the complete factored expression.

$$r^{2}\left(r+2\right)\left(r+7\right)$$

Evaluate

$$\left(r+2\right)\left(r+7\right)r^{2}$$