Algebraic Simplification Techniques#
Simplifying Expressions Involving Fractions#
Adding or Subtracting Fractions
If the fractions have a common denominator, then they can be combined into a single fraction by adding or subtracting the numerators accordingly.
e.g., \(\dfrac{2}{3} + \dfrac{5}{3} = \dfrac{7}{3}\)
If the fractions do not have a common denominator, then a common denominator can be formed by multiplying the denominators together.
e.g., \(\dfrac{2}{3} - \dfrac{5}{7} = \dfrac{2\cdot 7- 3\cdot5}{3\cdot 7} = -\dfrac{1}{21}\)
Multiplying Fractions
To multiply two fractions, multiply the numerators and multiply the denominators of the two fractions.
e.g., \(\dfrac{2}{3} \times \dfrac{5}{7} = \dfrac{2\cdot 5}{3\cdot 7} = \dfrac{10}{21}\)
Dividing Fractions
To divide two fractions, remember that division is the same as multiplication by the reciprocal. In other words, dividing by \(C/D\) is the same as multiplying by \(D/C\).
e.g., \(\dfrac{2}{3} \div \dfrac{5}{7} = \dfrac{2}{3} \times \dfrac{7}{5} = \dfrac{14}{15}\)
Example 1#
Subtracting fractions with the same denominator
Rewrite \(\dfrac{x+1}{x+4} - \dfrac{3x-2}{x+4}\) as a single ratio.
Step 1: Subtract fractions with common denominator.
Since the ratios already have the same denominator, we need only apply the formula for subtracting fractions with a common denominator.
Step 2: Apply the formula from Step 1.
Apply the formula from Step 1 with \(A = x+1\), \(B=x+4\), and \(C = 3x-2\).
Example 2#
Adding fractions with different denominators
Rewrite \(\dfrac{3}{x} + \dfrac{4}{x-5}\) as a single ratio.
Step 1: Apply formula for adding fractions with different denominators.
Since the ratios do not have the same denominator, we will apply the formula for adding fractions with different denominators.
Step 2: Apply the formula from Step 1.
Apply the formula from Step 1 with \(A = 3\), \(B=x\), \(C = 4\), and \(D = x-5\).
Example 3#
Simplifying quotients of fractions
Simplify
Step 1: Recall formula for the quotient of fractions.
Step 2: Use formula from Step 1 to rewrite equation.
Distributive Property of Multiplication#
Distributivity
The distributive property of multiplication can be used to rewrite a product (where at least one factor is a sum) as a sum.
e.g., \(7(3+2) = 7\cdot 3 + 7\cdot 2\)
Example 4#
Expanding a product of polynomials
Expand \(x^2(5x^3 + 7)\).
Step 1: Use distributivity to expand given product.
The FOIL Method#
The FOIL method is a way to remember how to apply the distributive property of multiplication when expanding the product of two binomial expressions. FOIL is an acronym for
F-O-I-L
First (i.e., multiply the first terms from each binomial)
Outer (i.e., multiply the first term from the first factor and the second term from the second factor)
Inner (i.e., multiply the second term from the first factor and the first term from the second factor)
Last (i.e., multiply the second terms of each binomial)
Example 5#
Applying the FOIL method
Expand \((x+2)(3x-5)\) using the FOIL method.
Step 1: Recall the formula for the FOIL method.
Step 2: Use the FOIL method to expand the expression.
Squaring a Binomial#
Applying the FOIL method to a binomial
When applying the FOIL method to the square of a binomial (i.e., \((a+b)^2\) or \((a-b)^2\)), we arrive at the following formulas:
Example 6#
Expanding a binomial
Expand \((3x-5)^2\) by squaring the binomial.
Step 1: Apply binomial formula.
Apply \((a - b)^2 = a^2 - 2ab + b^2\) with \(a=3x\) and \(b=5\).
Example 7#
Expanding a product of a monomial and a binomial
Expand \(3x^5(4+x)^2\).
Step 1: Apply binomial formula.
Apply \((a + b)^2 = a^2 + 2ab + b^2\) with \(a=4\) and \(b=x\).