Solve f^3+12f^2+35f

$$f ^ { 3 } + 12 f ^ { 2 } + 35 f$$

Factor

$$f\left(f+5\right)\left(f+7\right)$$

Solution Steps

Factor out f.

$$f\left(f^{2}+12f+35\right)$$

Consider f^{2}+12f+35. Factor the expression by grouping. First, the expression needs to be rewritten as f^{2}+af+bf+35. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\times 35=35$$

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 35.

$$1,35 5,7$$

Calculate the sum for each pair.

$$1+35=36 5+7=12$$

The solution is the pair that gives sum 12.

$$a=5 b=7$$

Rewrite f^{2}+12f+35 as \left(f^{2}+5f\right)+\left(7f+35\right).

$$\left(f^{2}+5f\right)+\left(7f+35\right)$$

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

$$f\left(f+5\right)+7\left(f+5\right)$$

Factor out common term f+5 by using distributive property.

$$\left(f+5\right)\left(f+7\right)$$

Rewrite the complete factored expression.

$$f\left(f+5\right)\left(f+7\right)$$

Evaluate

$$f\left(f+5\right)\left(f+7\right)$$