Solve f(x)=2x^3-7x^2+7x-2. | Microsoft Math Solver

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 2. One such root is 1. Factor the polynomial by dividing it by x-1.

a+b=-5 ab=2\times 2=4

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

-1,-4 -2,-2

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 4.

-1-4=-5 -2-2=-4

Calculate the sum for each pair.

a=-4 b=-1

The solution is the pair that gives sum -5.

\left(2x^{2}-4x\right)+\left(-x+2\right)

Rewrite 2x^{2}-5x+2 as \left(2x^{2}-4x\right)+\left(-x+2\right).

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

Factor out 2x in the first and -1 in the second group.

\left(x-2\right)\left(2x-1\right)

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