UPSKILL MATH PLUS

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Rational Expression
An expression is called a rational expression if it can be written in the form $$\frac{p(x)}{q(x)}$$ where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x) \neq 0$$.

A rational expression is the ratio of two polynomials.
Example:
The following are a few examples of rational expressions.

1. $$\frac{x + y}{x - y}$$, where $$x \neq y$$

2. $$\frac{3ab}{5a^2b^3}$$

3. $$\frac{x^2 + 3x}{x^5}$$
Reduction of Rational Expression
A rational expression $$\frac{p(x)}{q(x)}$$ is said to be in its lowest form if the greatest common divisor of $$p(x)$$ and $$q(x)$$ is $$1$$.

That is $$GCD \left(p(x), q(x) \right)$$ = $$1$$.
Working rule to reduce a rational expression to its lowest form:
Step 1: Simplify or factorise the numerator $$p(x)$$ and the denominator $$q(x)$$.

Step 2: Cancel out the common factors in the numerator and the denominator.

Step 3: The final expression obtained after the above two steps is the rational expression in its lowest form.
Example:
Reduce the expression $$\frac{x^2 + 5x + 6}{x + 2}$$.

Solution:

Step 1: Factorise the numerator $$x^2 + 5x + 6$$ by splitting the middle term.

$$x^2 + 5x + 6$$ $$=$$ $$x^2 + 2x + 3x + 6$$

$$=$$ $$x (x + 2) + 3 (x + 2)$$

$$=$$ $$(x + 2)(x + 3)$$

Step 2: Rewrite the expression and cancel out the common factors.

$$\frac{x^2 + 5x + 6}{x + 2}$$ $$=$$ $$\frac{(x + 2)(x + 3)}{x + 2}$$

$$=$$ $\frac{\overline{)\left(x+2\right)}\left(x+3\right)}{\overline{)\left(x+2\right)}}$

$$=$$ $$x + 3$$

Step 3: Write the rational expression in its lowest form.

$$\frac{x^2 + 5x + 6}{x + 2}$$ $$=$$ $$x + 3$$

Therefore, the rational expression $$\frac{x^2 + 5x + 6}{x + 2}$$ is reduced to $$x + 3$$.