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Algebraic Expressions (continued) Factoring Examples: Factor each expression −2x3 + 16x 2x2 y − 6xy 2 + 3xy To factor a polynomial of the form x2 + bx + c, we need to find two numbers that add up to b and multiply to c. x2 − 6x + 5 To factor a polynomial of the form ax2 + bx + c, we need to find factors (px + r) and (qx + s) such that ax2 + bx + c = (px + r)(qx + s) = pqx2 + (ps + qr)x + rs That means we look for numbers p, q, r and s such that pq = a, ps + qr = b and rs = c. 3x2 − 16x + 5 3(3y + 2)2 − 16(3y + 2) + 5 1 Special factoring formulas A2 − B 2 = (A − B)(A + B) A2 + 2AB + B 2 = (A + B)2 A2 − 2AB + B 2 = (A − B)2 A3 − B 3 = (A − B)(A2 + AB + B 2 ) A3 + B 3 = (A + B)(A2 − AB + B 2 ) Examples: 16z 2 − 24z + 9 1 + 1000y 3 More factoring examples: Factoring by grouping terms. x3 + x2 + x + 1 Factoring expressions with rational exponents. x−3/2 + 2x−1/2 + x1/2 Factor completely. y 4 (y + 2)3 + y 5 (y + 2)4 2 difference of squares perfect square perfect square difference of cubes sum of cubes Rational Expressions A quotient of two polynomials is called a rational expression. A quotient of two algebraic expression is called a fractional expression. √ x5 + 2 x2 − 1 x−4 + x3 x3/2 The domain of a quotient is the set of real numbers that the variable in the quotient is permitted to have, so that the quotient is defined. Examples: Find the domain of the following quotients. x2 + 1 x2 − x − 2 √ 2x x+1 How to handle fractional expressions: We use the laws of inverses of real numbers (Lecture 1) to simplify, multiply, divide, add or subtract fractional expressions. Examples: Perform the multiplication, division, addition or subtraction if needed and simplify. x2 − x − 12 x2 + 5x + 6 3 1 − x2 x3 − 1 x2 − 2x − 15 x + 3 · x2 − 9 x−5 x3 x+1 x x2 +2x+1 3 x − x−4 x+6 2 3 1 − + 2 2 x + 1 (x + 1) x −1 Simplify the following compound fraction . x y 1 x2 − − y x 1 y2 4 x−2 − y −2 x−1 + y −1 1 1+x+h − h 1 1+x Rationalizing the denominator (numerator) means to eliminate terms like √ both the numerA + C from the denominator (numerator) by multiplying √ ator and the denominator by the conjugate A − C. Examples: 1 √ 2− 3 √ √ x− x+h √ h+ x+h 5