Solve 10x^7+28x^6-6x^5

$$10 x ^ { 7 } + 28 x ^ { 6 } - 6 x ^ { 5 }$$

Factor

$$2\left(5x-1\right)\left(x+3\right)x^{5}$$

Solution Steps

Factor out 2.

$$2\left(5x^{7}+14x^{6}-3x^{5}\right)$$

Consider 5x^{7}+14x^{6}-3x^{5}. Factor out x^{5}.

$$x^{5}\left(5x^{2}+14x-3\right)$$

Consider 5x^{2}+14x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=14 ab=5\left(-3\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 14.

$$a=-1 b=15$$

Rewrite 5x^{2}+14x-3 as \left(5x^{2}-x\right)+\left(15x-3\right).

$$\left(5x^{2}-x\right)+\left(15x-3\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(5x-1\right)+3\left(5x-1\right)$$

Factor out common term 5x-1 by using distributive property.

$$\left(5x-1\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$2x^{5}\left(5x-1\right)\left(x+3\right)$$

Evaluate

$$2\left(5x-1\right)\left(x+3\right)x^{5}$$