Rational Expressions: Learn It 1

  • Simplify rational expressions
  • Practice how to multiply, divide, add, and subtract rational expressions

Rational Expressions

A pastry shop has fixed costs of [latex]\$280[/latex] per week and variable costs of [latex]\$9[/latex] per box of pastries. The shop’s costs per week in terms of [latex]x[/latex], the number of boxes made, is [latex]280+9x[/latex]. We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.

[latex]\Large{\dfrac{280+9x}{x}}[/latex]

Notice that the result is a polynomial expression divided by a second polynomial expression. This is known as a rational expression.

rational expressions

Rational expressions are formed when one polynomial is divided by another, resulting in a fraction-like form where the numerator and the denominator are both polynomials.

[latex]\dfrac{P(x)}{Q(x)}[/latex] where [latex]P(x)[/latex] and [latex]Q(x)[/latex] are polynomials.

The domain of a rational expression includes all real numbers except those that make the denominator equal to zero. When the denominator equals zero, the expression is undefined. This concept is rooted in the fundamental principle that division by zero is impossible in mathematics.

rational expressions and undefined values

A rational expression is undefined when its denominator equals zero.

How to: Find the undefined values of a rational expression

  1. Isolate the denominator.
  2. Set the denominator equal to zero.
  3. Solve the resulting equation.

The solutions to this equation are the values that make the rational expression undefined.

Determine the value(s) of [latex]x[/latex] for which the following rational expression is undefined:

[latex]\frac{x^2 - 4}{x - 2}[/latex]

Find the values of [latex]x[/latex] that make the following rational expression undefined:

[latex]\frac{x^2 + 3x}{x^2 - 4}[/latex]

Determine all values of [latex]x[/latex] for which the following rational expression is undefined:

[latex]\frac{2x^2 - 5}{x^3 - x}[/latex]

We can apply the properties of fractions to rational expressions such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

[latex]\dfrac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[/latex]

We can factor the numerator and denominator to rewrite the expression as [latex]\dfrac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}[/latex].

Then we can simplify the expression by canceling the common factor [latex]\left(x+4\right)[/latex] to get [latex]\dfrac{x+4}{x+7}[/latex].
How To: Given a rational expression, simplify it

  1. Factor the numerator and denominator.
  2. Cancel any common factors.
Simplify [latex]\dfrac{{x}^{2}-9}{{x}^{2}+4x+3}[/latex].

Can the [latex]{x}^{2}[/latex] term be cancelled in the above example?No. A factor is an expression that is multiplied by another expression. The term [latex]x^2[/latex] is part of the polynomial terms in both the numerator and denominator, but it is not a standalone factor.Polynomial expressions need to be fully factored, and only common binomial or monomial factors that appear as a product in both the numerator and denominator can be canceled. Canceling should only occur with factors, not terms that are part of a sum or difference in the polynomials, as altering these without proper factoring could change the value of the expression.
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