8.11: Summary of Key Concepts
- Page ID
- 60053
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Summary Of Key Concepts
Rational Expression:
A rational expression is an algebraic expression that can be written as the quotient of two polynomials. An example of a rational expression is:
\(\dfrac{x^2 + 3x - 1}{7x - 4}\)
Domain of a Rational Expression
The domain of a rational expression is the collection of values for which the rational expression is defined. These values can be found by determining the values that will not produce zero in the denominator of the expression.
The domain of \(\dfrac{x+6}{x+8}\) is the collection of all numbers except \(-8\).
Equality Property of Fractions
If \(\dfrac{a}{b} = \dfrac{c}{d}\), then \(ad = bc\).
If \(ad = bc\), then \(\dfrac{a}{b} = \dfrac{c}{d}\)
Negative Property of Fractions
\(\dfrac{-a}{b} = \dfrac{a}{-b} = -\dfrac{a}{b}\)
Reducing a Rational Expression
- Factor the numerator and denominator completely.
- Divide the numerator and denominator by any factors they have in common.
Common Cancelling Error
\(\dfrac{x + 54}{x + 7} \not = \dfrac{\cancel{x} + 4}{\cancel{x} + 7} \not = \dfrac{4}{7}\)
Since \(x\) is not a common factor, it cannot be cancelled.
Multiplying Rational Expressions
- Factor all numerators and denominators.
- Reduce to lowest terms first by dividing out all common factors.
- Multiply numerators together.
- Multiply denominators together.
It will be more convenient to leave the denominator in factored form.
Division of Rational Expressions
\(\dfrac{P}{Q} \div \dfrac{R}{S} = \dfrac{P}{Q} \cdot \dfrac{S}{R} = \dfrac{P \cdot S}{Q \cdot R}\)
Building Rational Expressions
\(\dfrac{P}{Q} \cdot \dfrac{b}{b} = \dfrac{Pb}{Qb}\)
Building rational expressions is exactly the opposite of reducing rational expressions. It is often useful in adding or subtracting rational expressions.
The building factor may be determined by dividing the original denominator into the new denominator. The quotient will be the building factor. It is this factor that will multiply the original numerator.
Least Common Denominator LCD
The LCD is the polynomial of least degree divisible by each denominator. It is found as follows:
- Factor each denominator. Use exponents for repeated factors.
- Write each different factor that appears. If a factor appears more than once, use only the factor with the highest exponent.
- The LCD is the product of the factors written in step 2.
Fundamental Rule for Adding or Subtracting Rational Expressions
To add or subtract rational expressions conveniently, they should have the same denominator.
Adding and Subtracting Rational Expressions
\(\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}\) and \(\dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a - b}{c}\)
Note that we combine only the numerators.
Rational Equation
A rational equation is a statement that two rational expressions are equal.
Clearing an Equation of Fractions
To clear an equation of fractions, multiply both sides of the equation by the LCD. This amounts to multiplying every term by the LCD.
Solving a Rational Equation
- Determine all values that must be excluded as solutions by finding the values that produce zero in the denominator.
- Clear the equation of fractions by multiplying every term by the LCD.
- Solve this nonfractional equation for the variable. Check to see if any of these potential solutions are excluded values.
- Check the solution by substitution.
Extraneous Solution
A potential solution that has been excluded because it creates an undefined expression (perhaps, division by zero) is called an extraneous solution.