## steve816 one year ago Is it possible to factor this without the use of a calculator?

1. steve816

$x^3+x^2-5x+3$

2. zepdrix

You can apply your Rational Root Theorem to look for "possible roots". The leading coefficient is a 1 on the x^3. Factors of 1 include 1 and -1. The constant on the end is a 3. Factors of 3 include $$\large\rm \color{orangered}{1, 3}$$ and $$\large\rm\color{orangered}{-1, -3}$$. Rational Root Theorem tells us to to take the combination of the constant factors and divide by the leading coefficient factors, individually. These will give us our possible roots which are rational. Here is the list for this specific polynomial:$\large\rm \pm\frac{\color{orangered}{1}}{1},\pm\frac{\color{orangered}{-1}}{1},\pm\frac{\color{orangered}{3}}{1},\pm\frac{\color{orangered}{-3}}{1}$I skipped any that would repeat.

3. zepdrix

So IF this polynomial has a rational root, it has to be $$\large\rm 1,-1,3\text{ or }-3$$.

4. zepdrix

Let's check x=1, nice simple process.$\large\rm x^3+x^2-5x+3\qquad\to\qquad (1)^3+(1)^2-5(1)+3$This results in 0. Good! We've found one of our roots / zeroes / solutions of this polynomial. You can apply polynomial long division or synthetic division to factor it down to a quadratic.

5. steve816

Wow, great explanation! Thanks for your time and effort :)