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Exponents and Polynomials The Rules of Exponents The Product Rule The Product Rule To multiply two exponential expressions that have the same base, keep the base and add the exponents. xa · xb = xa + b Example: Multiply. a.) c4 · c5 = c4 + 5 = c9 b.) 3a3 · a6 c.) 4w2 · 2w5 Numerical coefficient = 3a3 + 6 = 3a9 = (4)(2)w2 + 5 = 8w7 3 The Quotient Rule The Quotient Rule (Part 1) Use this form if the larger exponent is a a b x x in the numerator and x 0. b x Example: Divide. 6 a.) 3 362 34 2 3 b.) y 7 y 74 y 3 y4 Remember that the base does not change. 4 The Quotient Rule The Quotient Rule (Part 2) a Use this form if the larger exponent is x 1 b ba in the denominator and x 0. x x Example: Divide. 2 a.) 3 1 1 39 392 37 b.) z 1 1 z11 z116 z5 Remember that the base does not change. 6 5 The Quotient Rule The Quotient Rule (Part 3) xa x0 1 xa if x 0 (00 remains undefined). Example: Divide. a.) 1512 150 1 12 15 b.) a 3b0 a 32 (1) a 2 a 6 Raising Exponential Expressions to a Power Raising a Power to a Power To raise a power to a power, keep the same base and multiply the exponents. x a b x ab Example: Simplify. a.) (x5)3 = x5·3 = x15 b.) (y3)3 = y3·3 = y9 7 Product Raised to a Power Product Raised to a Power a a xy x y a Example: Simplify. a.) (2c)3 = (2)3c3 = 8c3 b.) (5xy)2 c.) (4x3y2)3 = (5)2(xy)2 = 25x2y2 = 43x9y6 = 64x9y6 8 Quotient Raised to a Power Quotient Raised to a Power a x xa y ya if y 0. Example: Simplify. 4 a.) x x4 y y 4 b.) 3a 3 a 81a12 2 8 2 4 b b b 3 4 4 3 4 9 Negative Exponents & Scientific Notation Negative Exponents Definition of a Negative Exponent x n 1n x if x 0. Example: Write with positive exponents. 4 14 h a.) h b.) 2a 3 5 1 2a 3 5 1 15 32a 11 Negative Exponents Properties of Negative Exponents 1 xn x n m x y n yn m x Example: Simplify. Write the expression with no negative exponents. 3 5 a.) 3 2 5 x y x x x 6 2 5 x y yy y b.) 4 2 3 2ab c 2 3 a 3b( 4)( 3) c (2)( 3) 12 b 3 3 12 6 2 a b c 2 3 a 3c 6 12 Scientific Notation Scientific Notation A number is written in scientific notation if it is in the form a × 10n, where 1 a 10 and n is an integer. 8200 = 8.2 1000 = 8.2 103 Greater than 1 and less than 10 Power of 10 34,200,000 = 3.42 10000000 = 3.42 107 Scientific notation 13 Decimal Notation to Scientific Notation Example: Write 67,300 in scientific notation. What power? 67,300. = 6.73 10n Starting position of decimal point Ending position of decimal point The decimal point was moved 4 places to the left, so we use a power of 4. 67,300 = 6.73 104 A number that is larger than 10 and written in scientific notation will always have a positive exponent as the power of 10. 14 Decimal Notation to Scientific Notation Example: Write 0.048 in scientific notation. What power? 0.048 = 4.8 10n Starting position of decimal point Ending position of decimal point The decimal point was moved 2 places to the right, so we use a power of –2. 0.048 = 4.8 10–2 A number that is smaller than 1 and written in scientific notation will always have a negative exponent as the power of 10. 15 Scientific Notation to Decimal Notation Example: Write 9.1 104 in decimal notation. 9.1 104 = 9.1000 104 = 91,000 Move the decimal point 4 places to the right. Example: Write 6.72 10–3 in decimal notation. 6.72 10–3 = 6.72 10–3 = 0.00672 Move the decimal point 3 places to the left. 16 Fundamental Polynomial Operations Recognizing Polynomials A polynomial in x is the sum of a finite number of terms of the form axn , where a is any real number and n is a whole number. 5x2 + x – 3, 1.2c3 + 5.1 A multivariable polynomial is a polynomial with more than one variable. 4y3z5 + 7w2 + 6, 2a3 + b2 – ab 18 Degrees of Terms & Polynomials The degree of a term is the sum of the exponents of all of the variables in the term. 5x2 The degree is 2. The degree is 3. 2c3 + 5.1 The degree of a polynomial is the highest degree of all of the terms in the polynomial. The degree is 8. 4y3z5 + 7w2 + 6 The degree is 3. 2a3 + b2 – ab 19 Types of Polynomials A monomial has one term. 5x2, a, 2c5d A binomial has two terms. 5x2 + 3, ab – 4 , 2 + c A trinomial has three terms. 5x2 + x – 3, ab – a + b , –c2 + c + 2 20 Adding Polynomials To add polynomials, add the like terms. Example: Add. (5x2 – x + 6) + (2x2 – 3) (5x2 – x + 6) + (2x2 – 3) = [5x2 + 2x2] + (– x ) + [6 + (– 3)] = 7x2 + (– x ) + 3 = 7x2 – x + 3 A polynomial is written in decreasing order when each exponent is decreasing. 21 Subtracting Polynomials To subtract two polynomials, change the sign of each term in the second polynomial and then add the like terms. Example: Subtract. (3a4 + 5a – 6) – (2a4 + 2a – 3) (3a4 + 5a – 6) – (2a4 + 2a – 3) = (3a4 + 5a – 6) + (– 2a4 – 2a + 3) = (3a4 – 2a4) + (5a – 2a) + (–6 + 3) = a4 + 3a – 3 22 Evaluating Polynomials Polynomials may be used to predict a value. This is done by evaluating the polynomial. Example: Evaluate the polynomial. Let a = 3. – 2a4 + 2a – 3 – 2(3)4 + 2(3) – 3 = – 2(81) + 6 – 3 = – 162 + 6 – 3 = – 159 23 Multiplying Polynomials Multiplying Polynomials Use the distributive property to multiply a monomial by a polynomial. Example: a.) Multiply. 3(c – 4) 3(c – 4) = 3(c) + 3(– 4) = 3c – 12 b.) Multiply. 2x3(x2 – x + 2) 2x3(x2 – x + 2) = 2x3(x2) + 2x3(– x) + 2x3(2) = 2x5 – 2x4 + 4x3 25 Multiplying Two Binomials To multiply two binomials, the distributive property is used so that every term in one polynomial is multiplied by every term in the other polynomial. Example: Multiply. (7x + 3)(2x + 4) (7x + 3)(2x + 4) = (7x + 3)(2x) + (7x + 3)(4) = 14x2 + 6x + 28x + 12 = 14x2 + 34x + 12 This method used to multiply two binomials is referred to as the FOIL method. 26 The FOIL Method Consider (a + b)(c + d): F “First” Multiply the first terms together. F (a + b) (c + d) O “Outer” Multiply the outer terms together. O (a + b) (c + d) I product ad “Inner” Multiply the inner terms together. I (a + b) (c + d) L product ac “Last” product bc Multiply the last terms together. L (a + b) (c + d) product bd The product of the two binomials is the sum of these four products: (a + b)(c + d) = ac + ad + bc + bd 27 The FOIL Method Example: Multiply using the FOIL method. (7x + 3)(2x + 4) L F (7x + 3)(2x + 4) I O F O I L = (7x)(2x) + (7x)(4) + (3)(2x) + (3)(4) = 14x 2 + 28x + 6x + 12 = 14x 2 + 34x + 12 28 The FOIL Method Example: Multiply. (3x + 4)(5x y) (3x + 4)(5x y) = (3x)(5x) + (3x)( y) + (4)(5x) + (4)( y) F O I L = 15x2 + ( 3xy) + 20x + ( 4y) = 15x2 3xy + 20x 4y 29 Multiplication: Special Cases Multiplying Binomials: (a + b)(a b) Example: Multiply. (2x + 4)(2x 4) (2x + 4)(2x 4) = (2x)(2x) + (2x)( 4) + (4)(2x) + (4)( 4) F O I L = 4x2 + ( 8x) + 8x + ( 16) The inner and outer products cancel. = 4x2 16 31 Multiplying Binomials: (a + b)(a b) Multiplying Binomials: A Sum and a Difference (a + b)(a b) = a2 – b2 Example: Multiply. (5a + 3)(5a 3) (5a + 3)(5a 3) = (5a)2 32 = 25a2 9 Example: Multiply. (8c + 2d)(8c 2d) (8c + 2d)(8c 2d) = (8c)2 (2d)2 = 64c2 4d2 32 Multiplying Binomials: (a + b)2 and (a b)2 Example: Multiply. (x + 6)2 (x + 6)2 = (x + 6)(x + 6) = (x)(x) + (x)(6) + (6)(x) + (6)(6) F O I L = x2 + 6x + 6x + 36 The inner and outer products are the same. = x2 + 12x + 36 33 Multiplying Binomials: (a + b)2 and (a b)2 Squaring Binomials (a + b)2 = a2 + 2ab + b2 (a b)2 = a2 – 2ab + b2 Example: Multiply. (12a 3)2 (12a 3)2 = (12a)2 2(12a)(3) + (3)2 = 144a2 72a + 9 Example: Multiply. (x + y)2 (x + y)2 = x2 + 2xy + y2 34 Multiplying Polynomials with More than One Term To multiply any two polynomials, vertical multiplication may be used. Example: Multiply. (w – 1)(2w2 + 7w + 3) 2w2 + 7w + 3 w–1 Keep the terms lined up. Write vertically, lining up the terms. – 2w2 – 7w – 3 2w3 + 7w2 + 3w Multiply – 1(2w2 + 7w + 3). 2w3 + 5w2 – 4w – 3 Add the terms in each column. Multiply w(2w2 + 7w + 3). 35 Multiplying Polynomials with More than One Term To multiply any two polynomials, the distributive property may also be used. Example: Multiply. (w – 1)(2w2 + 7w + 3) (w – 1)(2w2 + 7w + 3) Multiply the first term in the first polynomial by every term in the second polynomial… = w(2w2) + w(7w) + w(3) + (– 1)2w2 + (– 1)7w +(– 1)3 … and multiply the second term in the first polynomial by every term in the second polynomial. = 2w3 + 7w2 + 3w +(– 2w2) + (– 7w) + (– 3) = 2w3 + 5w2 – 4w – 3 36