Solve 10w^7-26w^6-12w^5

$$10 w ^ { 7 } - 26 w ^ { 6 } - 12 w ^ { 5 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(5w^{7}-13w^{6}-6w^{5}\right)$$

Consider 5w^{7}-13w^{6}-6w^{5}. Factor out w^{5}.

$$w^{5}\left(5w^{2}-13w-6\right)$$

Consider 5w^{2}-13w-6. Factor the expression by grouping. First, the expression needs to be rewritten as 5w^{2}+aw+bw-6. To find a and b, set up a system to be solved.

$$a+b=-13 ab=5\left(-6\right)=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -13.

$$a=-15 b=2$$

Rewrite 5w^{2}-13w-6 as \left(5w^{2}-15w\right)+\left(2w-6\right).

$$\left(5w^{2}-15w\right)+\left(2w-6\right)$$

Factor out 5w in the first and 2 in the second group.

$$5w\left(w-3\right)+2\left(w-3\right)$$

Factor out common term w-3 by using distributive property.

$$\left(w-3\right)\left(5w+2\right)$$

Rewrite the complete factored expression.

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

Evaluate

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