Question

How do you reduce to the lowest term $\dfrac{{{x^3} - 3x}}{{{x^2} - 3x}}?$

Answer 1

Hint: As we know that the above fraction consists of rational expressions in the numerator and denominator both. When the numerator and denominator consists of polynomials such fractions, then we call it rational expressions. As for example: $\dfrac{{4x + 2}}{{3x + 3}}$. We can convert these into lowest terms by arithmetic operations i.e. by taking the common factors out.

Complete step by step solution:
As per the question we have $\dfrac{{{x^3} - 3x}}{{{x^2} - 3x}}$.
We will take the common factor out as ${x^3} - 3x$ can be written as $x({x^2} - 3)$. Similarly the denominator ${x^2} - 3x$ can be written as $x(x - 3)$.
By substituting the values in the fraction we have $\dfrac{{x({x^2} - 3x)}}{{x(x - 3)}} = \dfrac{{{x^2} - 3x}}{{x - 3}}$.

Hence the lowest term is $\dfrac{{{x^2} - 3}}{{x - 3}}$.

Note: We should know that a fraction is said to be written in the simplest form if the numerator and the denominator will not have a common factor except $1$. So we have to simplify the above fraction as small as possible and based on this definition, the simplest form can be found. We should also note that the question is asked to be in simplest form, not to divide the polynomials with each other.