Solve 75x^3-7x^2+22x^4

$$75 x ^ { 3 } - 7 x ^ { 2 } + 22 x ^ { 4 }$$

Factor

$$\left(11x-1\right)\left(2x+7\right)x^{2}$$

Solution Steps

Factor out x^{2}.

$$x^{2}\left(75x-7+22x^{2}\right)$$

Consider 75x-7+22x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$22x^{2}+75x-7$$

Factor the expression by grouping. First, the expression needs to be rewritten as 22x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=75 ab=22\left(-7\right)=-154$$

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 -154.

$$-1,154 -2,77 -7,22 -11,14$$

Calculate the sum for each pair.

$$-1+154=153 -2+77=75 -7+22=15 -11+14=3$$

The solution is the pair that gives sum 75.

$$a=-2 b=77$$

Rewrite 22x^{2}+75x-7 as \left(22x^{2}-2x\right)+\left(77x-7\right).

$$\left(22x^{2}-2x\right)+\left(77x-7\right)$$

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

$$2x\left(11x-1\right)+7\left(11x-1\right)$$

Factor out common term 11x-1 by using distributive property.

$$\left(11x-1\right)\left(2x+7\right)$$

Rewrite the complete factored expression.

$$x^{2}\left(11x-1\right)\left(2x+7\right)$$

Evaluate

$$\left(11x-1\right)\left(2x+7\right)x^{2}$$