Question

Is it possible to factor this without the use of a calculator?

Answer 1

\[x^3+x^2-5x+3\]

Answer 2

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.

Answer 3

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

Answer 4

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.

Answer 5

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