What is a rational expression?
Rational expression is a fraction with one variable – This is a fractional algebraic expression containing one variable. This variable is represented by a letter, such as x, and it appears in the numerator and/or denominator of the fraction. Essentially, it is a fraction, but more complex than simple numerical fractions, as its numerator or denominator may include variables, multiplication, addition, or subtraction.
Example:
Let’s analyse an example. Below you see a rational expression:
- x2 + 3xx2 – 9
As you can see, this is an algebraic expression with one unknown x, which is both in the numerator and denominator. The value of the fraction depends on the value of the variable x.
Important!
The fraction is meaningless if the denominator equals 0. This is because division by 0 is not possible.
Let’s explore steps to be followed in simplifying rational expressions?
First and foremost, it is important to understand why it is necessary to know how to simplify rational expressions.
- Clarity and easier understanding: Simplified rational expressions are easier to understand and are clearer in general. Complex expressions can hide important properties and relationships that become obvious only after simplifying these rational expressions.
- Easier problem solving: Simplified rational expressions often make it easier to solve equations or inequalities, as simplification eliminates unnecessary parts of the expression, facilitating arithmetic operations.
Below is an explanation of how to simplifyrational expressions:
- Factorization: First, if possible, factor the expressions in the numerator and denominator. The quick multiplication formulas will be very helpful here. This means finding the factors of both the numerator and the denominator, whose product gives the original expression.
- Simplification: Check if the numerator and denominator have the same or similar factors. If they do, both the numerator and denominator are divided by that factor (or those factors).
Let’s examine a few examples:
Example 1
Given a rational expression: x2 – 4x – 2. Let’s simplify it.
Can we factorize both the numerator and the denominator?
- The algebraic expression in the numerator is a difference of two squares, i.e., x2 – 4. Applying the quick multiplication formulas, you can factorize this expression into the product of two factors, i.e., (x – 2)(x + 2).
After this operation, we have the following expression:
- (x – 2)(x + 2)x – 2.
We see that both the numerator and denominator have the same factor x – 2. We divide both the numerator and denominator by this factor and simplify the fraction.
We obtain this simplified expression: x + 2
Example 2
Given rational expression: x2 – 9x2 – 6x + 9. Let’s simplify it.
Can we factorize both the numerator and the denominator?
- The algebraic expression in the numerator is a difference of two squares, i.e., x2 – 9. Applying the quick multiplication formulas, you can factorize this expression into the product of two factors, i.e., (x – 3)(x + 3).
- The expression in the denominator can also be factorized (apply the quick multiplication formulas – in this case, (a – b)² = a² – 2ab + b²). Applying this formula to the algebraic expression in the denominator, we get: x2 – 6x + 9 = (x – 3)2 or (x – 3)(x – 3).
After these operations, we have the following fraction:
- (x – 3)(x + 3)(x – 3)(x – 3)
We see that both the numerator and denominator have the same factor x – 3. We divide both the numerator and denominator by this factor and simplify the fraction.
We obtain this simplified rational expression: x + 3x – 3.
Important!
The values of x cannot be such that they make the denominator of the fraction equal to 0. In this case, x cannot be equal to 3.
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