Solve 10u^7-15u^6-25u^5

$$10 u ^ { 7 } - 15 u ^ { 6 } - 25 u ^ { 5 }$$

Factor

$$5\left(2u-5\right)\left(u+1\right)u^{5}$$

Solution Steps

Factor out 5.

$$5\left(2u^{7}-3u^{6}-5u^{5}\right)$$

Consider 2u^{7}-3u^{6}-5u^{5}. Factor out u^{5}.

$$u^{5}\left(2u^{2}-3u-5\right)$$

Consider 2u^{2}-3u-5. Factor the expression by grouping. First, the expression needs to be rewritten as 2u^{2}+au+bu-5. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-5\right)=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -3.

$$a=-5 b=2$$

Rewrite 2u^{2}-3u-5 as \left(2u^{2}-5u\right)+\left(2u-5\right).

$$\left(2u^{2}-5u\right)+\left(2u-5\right)$$

Factor out u in 2u^{2}-5u.

$$u\left(2u-5\right)+2u-5$$

Factor out common term 2u-5 by using distributive property.

$$\left(2u-5\right)\left(u+1\right)$$

Rewrite the complete factored expression.

$$5u^{5}\left(2u-5\right)\left(u+1\right)$$

Evaluate

$$5\left(2u-5\right)\left(u+1\right)u^{5}$$