How can I factorize this: "$X^3 + X^2 + X - 3$"

Question

I am going to elementary school & I am living in one of those deprived areas of Africa.

I can solve mathematical questions like this:

$$X^3 + X^2 + X +1 = X^2(X+1)+(X+1) = (X+1)(X^2+1)$$

Or even

\begin{align}X^2 − 2X + X^2 - X + 1 &= (X^2 - 2X + 1) + (X^2 - X) \\ &= (X - 1)^2 + X(X - 1) \\ &= (X-1)(X-1+X) \\ &= (X - 1)(2X - 1) \end{align}

But for a few months I have not been able to find a teacher around here who can factorize this:

$$X^3 + X^2 + X - 3$$

Do we have to solve it in this way?

$$X^3 + X^2 + X - 3 = X^2(X + 1) + X - 3$$

Or something else? I'd appreciate your help with this.

Answer 1

Hint: $(x^3+x^2+x-3):(x-1)=x^2+2x+3$

Answer 2

Surely $1$ is a root of $x^3+x^2+x-3$ therefore $x-1$ is a factor of it. We have $$x^3+x^2+x-3=(x-1)(x^2+x+1+x+1+1)=(x-1)(x^2+2x+3)$$

Answer 3

For this kind of problem it is worth knowing about the factor theorem

This says if $a$ is a root of your polynomial $f(x)$ i.e. $f(a)=0$ then $x-a$ is a factor of, i.e. divides, the polynomial.

In these kinds of problems it is worth trying a few values such as $\pm1, \pm2$.

In your example $f(1)=0$ so $x-1$ divides your polynomial, allowing you to factorize it as $(x-1)(Ax^2+Bx+C)$ where you need to find $A,B,C$.