Solve 60x^2-32x^3+x^4

$$60 x ^ { 2 } - 32 x ^ { 3 } + x ^ { 4 }$$

Factor

$$\left(x-30\right)\left(x-2\right)x^{2}$$

Solution Steps

Factor out x^{2}.

$$x^{2}\left(60-32x+x^{2}\right)$$

Consider 60-32x+x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-32x+60$$

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+60. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\times 60=60$$

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 60.

$$-1,-60 -2,-30 -3,-20 -4,-15 -5,-12 -6,-10$$

Calculate the sum for each pair.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

The solution is the pair that gives sum -32.

$$a=-30 b=-2$$

Rewrite x^{2}-32x+60 as \left(x^{2}-30x\right)+\left(-2x+60\right).

$$\left(x^{2}-30x\right)+\left(-2x+60\right)$$

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

$$x\left(x-30\right)-2\left(x-30\right)$$

Factor out common term x-30 by using distributive property.

$$\left(x-30\right)\left(x-2\right)$$

Rewrite the complete factored expression.

$$x^{2}\left(x-30\right)\left(x-2\right)$$

Evaluate

$$\left(x-30\right)\left(x-2\right)x^{2}$$