Question

How do you factor ${x^3} - 3x + 2$ ?

Answer 1

Hint: In the above question you were asked to factor ${x^3} - 3x + 2$. For solving this problem, you can use synthesis division. Synthetic division is a method of polynomial division in the special case of dividing by a linear factor. So, let us see how we can solve this problem.

Complete step by step solution:
Given in the question that we have to factor ${x^3} - 3x + 2$. You can easily observe that on putting x = 1, we get ${x^3} - 3x + 2 =$ . So (x-1) is a factor.
On using synthetic division for finding the remaining factor we get,
 $\Rightarrow {{x}^{3}}-3x+2=(x-1)({{x}^{2}}+x-2)$
Here also, you can observe that x = 1 gives us ${x^2} + x - 2 = 0$.
So now we have another (x-1) factor
 $\Rightarrow {{x}^{2}}+x-2=(x-1)(x+2)$

Therefore, the factor of ${x^3} - 3x + 2$ is $(x - 1)(x - 1)(x + 2)$.

Note:
In the above solution we get 3 factors because after applying synthetic division for the first time we get $(x - 1)({x^2} + x - 2)$ . $({x^2} + x - 2)$ can still be factored and that’s why we factored it again but only $({x^2} + x - 2)$ part. Also, we used synthetic division in our solution.