Solve 2x^6-3x^5-35x^4-2x^2+3x+35

$$2 x ^ { 6 } - 3 x ^ { 5 } - 35 x ^ { 4 } - 2 x ^ { 2 } + 3 x + 35$$

Factor

$$\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(2x+7\right)\left(x^{2}+1\right)$$

Solution Steps

Do the grouping 2x^{6}-3x^{5}-35x^{4}-2x^{2}+3x+35=\left(2x^{6}-3x^{5}-35x^{4}\right)+\left(-2x^{2}+3x+35\right), and factor out x^{4} in the first and -1 in the second group.

$$x^{4}\left(2x^{2}-3x-35\right)-\left(2x^{2}-3x-35\right)$$

Factor out common term 2x^{2}-3x-35 by using distributive property.

$$\left(2x^{2}-3x-35\right)\left(x^{4}-1\right)$$

Consider 2x^{2}-3x-35. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-35. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-35\right)=-70$$

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 -70.

$$1,-70 2,-35 5,-14 7,-10$$

Calculate the sum for each pair.

$$1-70=-69 2-35=-33 5-14=-9 7-10=-3$$

The solution is the pair that gives sum -3.

$$a=-10 b=7$$

Rewrite 2x^{2}-3x-35 as \left(2x^{2}-10x\right)+\left(7x-35\right).

$$\left(2x^{2}-10x\right)+\left(7x-35\right)$$

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

$$2x\left(x-5\right)+7\left(x-5\right)$$

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

$$\left(x-5\right)\left(2x+7\right)$$

Consider x^{4}-1. Rewrite x^{4}-1 as \left(x^{2}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Rewrite the complete factored expression. Polynomial x^{2}+1 is not factored since it does not have any rational roots.

$$\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(2x+7\right)$$

Evaluate

$$\left(x-5\right)\left(2x+7\right)\left(x^{4}-1\right)$$