8.10 Summary of key concepts

Summary of key concepts

Rational expression ( [link] )

Domain of a rational expression ( [link] )

Equality property of fraction ( [link] )

1. If $\frac{a}{b}=\frac{c}{d}$ , then $ad=bc$ .
2. If $ad=bc$ , then $\frac{a}{b}=\frac{c}{d}$ .

Negative property of fractions ( [link] )

Reducing a rational expression ( [link] )

1. Factor the numerator and denominator completely.
2. Divide the numerator and denominator by any factors they have in common.

Common cancelling error ( [link] )

Multiplying rational expressions ( [link] )

1. Factor all numerators and denominators.
2. Reduce to lowest terms first by dividing out all common factors.
3. Multiply numerators together.
4. Multiply denominators together.

Division of rational expressions ( [link] )

Building rational expressions ( [link] )

Least common denominator lcd ( [link] )

1. Factor each denominator. Use exponents for repeated factors.
2. Write each different factor that appears. If a factor appears more than once, use only the factor with the highest exponent.
3. The LCD is the product of the factors written in step 2.

Fundamental rule for adding or subtracting rational expressions ( [link] )

Adding and subtracting rational expressions ( [link] )

Rational equation ( [link] )

Clearing an equation of fractions ( [link] )

Solving a rational equation ( [link] )

1. Determine all values that must be excluded as solutions by finding the values that produce zero in the denominator.
2. Clear the equation of fractions by multiplying every term by the LCD.
3. Solve this nonfractional equation for the variable. Check to see if any of these potential solutions are excluded values.
4. Check the solution by substitution.

Extraneous solution ( [link] )