Simplifying Rational Expressions

Simplifying Rational Expressions Reducing Fraction Review  Fractions are in simplest form when the numerator & denominator have no common factors except 1.  Provided below are 4 methods for reducing fractions.

Method 1: Divisibility Rules  Use divisibility rules to divide both numerator and denominator.  Example: 18 18  2 9 both are divisible by 2 so , = 24 24  2 12 9 9  3 3 Can be reduced further? Yes, divide by 3 so = 12 12  3 4 3 Answer: 4

Method 2: Power of 10  If both numerator and denominator are a power of 10, divide by power of 10.  Example: 60 6 both are powers of 10 so drop 1 “0”, 100 10 6 6  2 3 Can be reduced further? Yes, divide by 2 so = 10 10  2 5 3 Answer: 5

Method 3: Find GCF  Find the greatest common factor of both numerator & denominator and divide by it. 18  Example: 24 Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 GCF: 6 18 18  6 3 Divide by 6 so = 24 24  6 4 3 Answer: 4

Method 4: Reduce by common primes  Write numerators & denominators as products of prime & reduce by common primes. 18  Example: 24 Prime Factorization of 18: 2 x 3 x 3 Prime Factorization of 24: 2 x 2 x 2 x 3 3 Answer: 4

1 Simplifying Rational Expressions Simplifying Algebraic Terms Review  When given an algebraic expression with numbers and letters in numerators & denominators, the following apply: o Reduce the number portion as you would in a fraction. See above for examples. o When dealing with the same base letter, simply subtract the exponents.

6x4  Example: 18x2 Step 1: Reduce the numbers 6 6  6 1 Can be reduced by 6 so = 18 18  6 3

Step 2: Reduce the variables x4 Same base of x so x4-2 = x2 x2

1x2 Answer: 3

Practice: Simplifying Algebraic Terms 15a2 32h3 12n4 1. 2. 3. 25a4 48h2 21n6

Factoring Quadratic Equations Review Provided below is brief summary of factoring quadratic equations. Refer to Quadratics notes for more details.

Factoring ax 2 – c 2 Factoring out GCF  Result is in the form: (√ax2 - √c2)(√ax2 + √c2)  Look for GCF and factor out  Examples:  Examples: o x2 - 25 = (x - 5)(x + 5) o x2 – 5x = x(x – 5) o 100 - x2 = (10 - x)(10 + x) o 4x – 4y = 4(x – y) o x3 – x2 = x2 (x + 1)

Practice: Factoring 1. 3x – 6 2. x2 – 2x 3. x2 – y2 4. 2x2 – 8

5. x2 + 3x + 2 6. x2 + 13x + 12 7. x2 + 2x - 15 8. x2 – 7x + 12

2 Simplifying Rational Expressions Rational Expressions:  Expression with variable in denominator 1 x 2 x2  5  Examples: x x  3 x2 10x 25

Simplifying Rational Expressions:  Remember how to simplify or reduce fractions with arithmetic and exponents.  Think of the expression in the numerator as 1 item and do the same for the denominator.  Make sure to identify any restriction values the denominator may have. o Examples: 1 Numerator: 1, denominator x x x 2 Numerator x+ 2, denominator x-3 x  3 x2  5 Numerator: x2 –5, denominator: x2 – 10x + 25 x2 10x 25  Factor out numerator & denominator and then look for things in common to eliminate.

Examples Simplifying Rational Expressions 3x2 1. Simplify 15xy 3 1 x2 x21 Step 1: Simplify the fraction = Step 2: Simplify the variables = 15 5 xy y x Answer: 5y x2 x 2. Simplify x2 4x 3

Step 1: Factor out numerator Step 2: Factor out denominator x2 + x = x(x + 1) x2 + 4x + 3 = (x + 1)(x+3) Restrictions: x  -1; x  -3

Step 3: Write factored out numerator over denominator & eliminate common terms x2 x x (x 1) x (x 1) = Since both have (x+1) cancel it out x2 4x 3 (x 3)(x 1) (x 3)(x 1) x = ; x  -1; x  -3 (x 3) Can this be further simplified? No because the numerator is x and the denominator is (x+3)

Practice Simplifying Rational Expressions Give all restrictions and simplify the following 21x2 y 32x3 y2 x2 y2 x2 5x 6 1. 2. 3. 4. 14xy2 24xy4 x y x2 x  6