Solve 22x^7-83x^6+21x^5

$$22 x ^ { 7 } - 83 x ^ { 6 } + 21 x ^ { 5 }$$

Factor

$$\left(2x-7\right)\left(11x-3\right)x^{5}$$

Solution Steps

Factor out x^{5}.

$$x^{5}\left(22x^{2}-83x+21\right)$$

Consider 22x^{2}-83x+21. Factor the expression by grouping. First, the expression needs to be rewritten as 22x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=-83 ab=22\times 21=462$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 462.

$$-1,-462 -2,-231 -3,-154 -6,-77 -7,-66 -11,-42 -14,-33 -21,-22$$

Calculate the sum for each pair.

$$-1-462=-463 -2-231=-233 -3-154=-157 -6-77=-83 -7-66=-73 -11-42=-53 -14-33=-47 -21-22=-43$$

The solution is the pair that gives sum -83.

$$a=-77 b=-6$$

Rewrite 22x^{2}-83x+21 as \left(22x^{2}-77x\right)+\left(-6x+21\right).

$$\left(22x^{2}-77x\right)+\left(-6x+21\right)$$

Factor out 11x in the first and -3 in the second group.

$$11x\left(2x-7\right)-3\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(11x-3\right)$$

Rewrite the complete factored expression.

$$x^{5}\left(2x-7\right)\left(11x-3\right)$$

Evaluate

$$\left(2x-7\right)\left(11x-3\right)x^{5}$$