<p> 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. </p><p>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</p><p>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</p><p>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</p><p>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</p><p>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. </p><p>6x4  Example: 18x2 Step 1: Reduce the numbers 6 6  6 1 Can be reduced by 6 so = 18 18  6 3</p><p>Step 2: Reduce the variables x4 Same base of x so x4-2 = x2 x2</p><p>1x2 Answer: 3</p><p>Practice: Simplifying Algebraic Terms 15a2 32h3 12n4 1. 2. 3. 25a4 48h2 21n6</p><p>Factoring Quadratic Equations Review Provided below is brief summary of factoring quadratic equations. Refer to Quadratics notes for more details.</p><p>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)</p><p>Practice: Factoring 1. 3x – 6 2. x2 – 2x 3. x2 – y2 4. 2x2 – 8</p><p>5. x2 + 3x + 2 6. x2 + 13x + 12 7. x2 + 2x - 15 8. x2 – 7x + 12</p><p>2 Simplifying Rational Expressions Rational Expressions:  Expression with variable in denominator 1 x 2 x2  5  Examples: x x  3 x2 10x 25</p><p>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.</p><p>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</p><p>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</p><p>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)</p><p>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</p><p>3</p>