Solve 30x^3y+115x^2y+35xy

$$30 x ^ { 3 } y + 115 x ^ { 2 } y + 35 x y$$

Factor

$$5xy\left(2x+7\right)\left(3x+1\right)$$

Solution Steps

Factor out 5.

$$5\left(6x^{3}y+23x^{2}y+7xy\right)$$

Consider 6x^{3}y+23x^{2}y+7xy. Factor out xy.

$$xy\left(6x^{2}+23x+7\right)$$

Consider 6x^{2}+23x+7. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=23 ab=6\times 7=42$$

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 42.

$$1,42 2,21 3,14 6,7$$

Calculate the sum for each pair.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

The solution is the pair that gives sum 23.

$$a=2 b=21$$

Rewrite 6x^{2}+23x+7 as \left(6x^{2}+2x\right)+\left(21x+7\right).

$$\left(6x^{2}+2x\right)+\left(21x+7\right)$$

Factor out 2x in the first and 7 in the second group.

$$2x\left(3x+1\right)+7\left(3x+1\right)$$

Factor out common term 3x+1 by using distributive property.

$$\left(3x+1\right)\left(2x+7\right)$$

Rewrite the complete factored expression.

$$5xy\left(3x+1\right)\left(2x+7\right)$$

Evaluate

$$5xy\left(2x+7\right)\left(3x+1\right)$$