Section 8.3 Adding and Subtracting Rational Expressions Like Denominators Unlike Denominators Complex Fractions

8.3 Adding and Subtracting Rational Expressions Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with LIKE DENOMINATORS, add or subtract the numerators (combine like terms) and use the SAME DENOMINATOR.

Examples – Add or Subtract Examples – Add or Subtract. Identify any x-values for which the expression is undefined. 1. 2.

Examples – Add or Subtract Examples – Add or Subtract. Identify any x-values for which the expression is undefined. 3. 4.

Examples – Add or Subtract Examples – Add or Subtract. Identify any x-values for which the expression is undefined. 5. 6.

Bell Ringer: Add or Subtract then Simplify 1. 2. 3. 4. 5.

Section 8.3 Adding and Subtracting Rational Expressions Like Denominators Unlike Denominators Complex Fractions

8.3 Adding and Subtracting Rational Expressions What is a "Multiple" ? The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Here are some examples: The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ... The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc... What is a "Common Multiple" ? When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of those numbers. For example, when you write down the multiples of 4 and 5, the common multiples are those that are found in both lists: The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,... Notice that 20 and 40 appear in both lists? So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too) What is the "Least Common Multiple" ? It is simply the smallest of the common multiples. In our previous example, the smallest of the common multiples is 20 ... ... so the Least Common Multiple of 4 and 5 is 20.

Least Common Multiples What is a Least Common Multiple?? The smallest (non-zero) number or term that is a multiple of two or more numbers or terms. How do I find it?? What about variables and exponents?? - use the highest exponent given…

Least Common Multiples Find the least common multiple for each pair… 2 and 8 3 and 7 6 and 10 2x and x2 4x2y3 and 6x4y5 x2 – 2x – 3 and x2 – x – 6

You Try… 4x3y7 and 3x5y4 (y – 2)(y + 5) and (y + 5)(y – 4) x2 – 4 and x2 + 5x + 6 x2 – 2x – 15 and x2 + 7x + 12

Least Common Denominators To add or subtract rational expressions with unlike denominators, first find the least common denominator (LCD). The LCD is the least common multiple of the polynomials in the denominators. To find the Least Common Multiple (LCM) of polynomials: Factor each polynomial completely. List the different factors. If the polynomials have common factors, use the highest power of each common factor.

To add rational expressions with unlike denominators, rewrite both expressions with the least common denominator (LCD). This process is similar to adding fractions…

ADD. Identify any x-values for which the expression in undefined. Example 7 Example 8

ADD. Identify any x-values for which the expression in undefined. Example 9 Example 10

SUBTRACT. Identify any x-values for which the expression in undefined. Example 11 Example 12

Section 8.3 Adding and Subtracting Rational Expressions Like Denominators Unlike Denominators Complex Fractions

8.3 Adding and Subtracting Rational Expressions Some rational expressions are complex fractions. A complex fraction contains one or more fractions in its numerator, its denominator, or both. Examples of complex fractions are shown below. Recall that the bar in a fraction represents division. Therefore, you can rewrite a complex fraction as a division problem and then simplify. You can also simplify complex fractions by using the LCD of the fractions in the numerator and denominator.

Simplify. Assume that all expressions are defined. Example 13 Example 14 x + 2 x – 1 x2 – 4 x2 – 1 6x x2 – 1 x x – 1

Simplify. Assume that all expressions are defined. Example 15 Example 16 4x + 8 6x2 3 2x 1 + x – 1 x 2 3 +

Homework: p. 588 (#13-15, 28-29) Bonus: #30