# Operations on Rational Expressions. Rational expressions are fractions in which the numerator and denominator are polynomials and the denominator does.

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Operations on Rational Expressions

Rational expressions are fractions in which the numerator and denominator are polynomials and the denominator does not equal zero. Example: Simplify, x – 3  0, x  3

1. Factor the numerator and denominator of each fraction. 2. Multiply the numerators and denominators of each fraction. 4. Write the answer in simplest form. 3. Divide by the common factors. To multiply rational expressions:

Factor the numerator and denominator of each fraction. Divide by common factors. Write the answer in simplest form. MultiplyExample: Multiply.

1. Multiply the dividend by the reciprocal of the divisor. The reciprocal of is. 2. Multiply the numerators. Then multiply the denominators. 4. Write the answer in simplest form. 3. Divide by the common factors. To divide rational expressions:

Example: Divide Multiply by the reciprocal of the divisor. Factor and multiply. Divide by the common factors. Simplest form.

The least common multiple (LCM) of two or more numbers is the least number that contains the prime factorization of each number. Examples: 1. Find the LCM of 10 and 4. 2. Find the LCM of 4x 2 + 4x and x 2 + 2x + 1. 4x 2 + 4x = (4x)(x +1) = 2 2 x (x + 1) x 2 + 2x + 1 = (x +1)(x +1) LCM = 2 2 x (x +1)(x +1) factors of 4x 2 + 4x factors of x 2 + 2x + 1 10 = (5 2) LCM = 2 2 5 factors of 10 factors of 4 4 = (2 2) = 4x 3 + 8x 2 + 4x = 20

LCM Fractions can be expressed in terms of the least common multiple of their denominators. Example: Write the fractions and in terms of the LCM of the denominators. The LCM of the denominators is 12x 2 (x – 2).

1.If necessary, rewrite the fractions with a common denominator. To add rational expressions: To subtract rational expressions: 2. Add the numerators of each fraction. **Denominator stays the same! 1.If necessary, rewrite the fractions with a common denominator. 2. Subtract the numerators of each fraction. **Denominator stays the same!

Example: Add Example: Subtract

Two rational expressions with different denominators can be added or subtracted after they are rewritten with a common denominator. Example: Add

Example: Subtract Add numerators. Factor. Divide. Simplest form.

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