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Chapter 12 Exponents and Polynomials Chapter Sections 12.1 – Exponents 12.2 – Negative Exponents and Scientific Notation 12.3 – Introduction to Polynomials 12.4 – Adding and Subtracting Polynomials 12.5 – Multiplying Polynomials 12.6 – Special Products 12.7 – Dividing Polynomials Martin-Gay, Developmental Mathematics 2 § 12.1 Exponents Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3 • 3 • 3 • 3 3 is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations. Martin-Gay, Developmental Mathematics 4 Evaluating Exponential Expressions Example Evaluate each of the following expressions. 34 = 3 • 3 • 3 • 3 = 81 (–5)2 = (– 5)(–5) = 25 –62 = – (6)(6) = –36 (2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512 3 • 42 = 3 • 4 • 4 = 48 Martin-Gay, Developmental Mathematics 5 Evaluating Exponential Expressions Example Evaluate each of the following expressions. a.) Find 3x2 when x = 5. 3x2 = 3(5)2 = 3(5 · 5) = 3 · 25 = 75 b.) Find –2x2 when x = –1. –2x2 = –2(–1)2 = –2(–1)(–1) = –2(1) = –2 Martin-Gay, Developmental Mathematics 6 The Product Rule Product Rule (applies to common bases only) am · an = am+n Example Simplify each of the following expressions. 32 · 34 = 32+4 = 36 = 3 · 3 · 3 · 3 · 3 · 3= 729 x4 · x5 = x4+5 = x9 z3 · z2 · z5= z3+2+5 = z10 (3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6 Martin-Gay, Developmental Mathematics 7 The Power Rule Power Rule (am)n = amn Example Simplify each of the following expressions. (23)3 = 23·3 = 29 = 512 (x4)2 = x4·2 = x8 Martin-Gay, Developmental Mathematics 8 The Power of a Product Rule Power of a Product Rule (ab)n = an · bn Example Simplify (5x2y)3. (5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3 Martin-Gay, Developmental Mathematics 9 The Power of a Quotient Rule Power of a Quotient Rule a b Example Simplify n a n b n a b n a n b n (5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3 Martin-Gay, Developmental Mathematics 10 The Power of a Quotient Rule Power of a Quotient Rule a b n a n b n Example Simplify the following expression. p 3 3r 2 4 2 4 p 3 3r 4 p 3 r 2 4 4 3 4 (Power of product rule) p 8 81r 12 (Power rule) Martin-Gay, Developmental Mathematics 11 The Quotient Rule Quotient Rule (applies to common bases only) a m a n a mn a0 Example Simplify the following expression. 4 9a b 3 ab 7 2 4 7 9 a b 2 3 ( a 4 1 )( b 7 2 ) 3 a 3 b 5 3 a b Group common bases together Martin-Gay, Developmental Mathematics 12 Zero Exponent Zero exponent a0 = 1, a 0 Note: 00 is undefined. Example Simplify each of the following expressions. 50 = 1 (xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1 –x0 = –(x0) = – 1 Martin-Gay, Developmental Mathematics 13 § 12.2 Negative Exponents and Scientific Notation Negative Exponents Using the quotient rule from section 3.1, x 4 x 6 x 46 x 2 x0 But what does x -2 mean? x 4 x 6 xxxx xxxxxx 1 xx Martin-Gay, Developmental Mathematics 1 x 2 15 Negative Exponents So, in order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows. If a 0, and n is an integer, then a n 1 a n Martin-Gay, Developmental Mathematics 16 Simplifying Expressions Example Simplify by writing each of the following expressions with positive exponents or calculating. 3 x 1 2 3 7 2x 9 1 x 4 2 1 7 2 x 4 Remember that without parentheses, x is the base for the exponent –4, not 2x Martin-Gay, Developmental Mathematics 17 Simplifying Expressions Example Simplify by writing each of the following expressions with positive exponents or calculating. x 3 3 1 x 2 1 3 ( 3) 2 3 2 1 9 1 ( 3) 2 1 9 Notice the difference in results when the parentheses are included around 3. Martin-Gay, Developmental Mathematics 18 Simplifying Expressions Example Simplify by writing each of the following expressions with positive exponents. 1 1) x 3 1 1 x x 2) y 2 4 x 3 x 3 1 3 1 2 x 1 y 4 y 4 x 2 (Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from a numerator to a denominator, or vice versa, and switch the exponent to its positive value.) Martin-Gay, Developmental Mathematics 19 Summary of Exponent Rules If m and n are integers and a and b are real numbers, then: Product Rule for exponents am · an = am+n Power Rule for exponents (am)n = amn Power of a Product (ab)n = an · bn n Power of a Quotient n a a , b0 n b b Quotient Rule for exponents a m a n a mn , a0 Zero exponent a0 = 1, a 0 Negative exponent a n 1 a n , a0 Martin-Gay, Developmental Mathematics 20 Simplifying Expressions Simplify by writing the following expression with positive exponents or calculating. 2 3 a b 4 7 3 3 a b 3 2 3 3 2 4 3 a b 7 a b 3 a b 3 a b 2 2 2 4 2 3 2 Power of a quotient rule 4 6 3 a b 8 3 a 14 b 2 6 4 14 3 a b 8 6 3 a b Power rule for exponents 3 2 2 7 2 3 2 Power of a product rule 2 6 3 4 8 a 14 6 b 26 4 3 a b 8 4 a 4 8 3 b 4 a 8 81b 4 Quotient rule for exponents Negative exponents Martin-Gay, Developmental Mathematics Negative exponents 21 Scientific Notation In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers. A positive number is written in scientific notation if it is written as a product of a number a, where 1 a < 10, and an integer power r of 10. a 10r Martin-Gay, Developmental Mathematics 22 Scientific Notation To Write a Number in Scientific Notation 1) Move the decimal point in the original number to the left or right, so that the new number has a value between 1 and 10. 2) Count the number of decimal places the decimal point is moved in Step 1. • If the original number is 10 or greater, the count is positive. • If the original number is less than 1, the count is negative. 3) Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2. Martin-Gay, Developmental Mathematics 23 Scientific Notation Example Write each of the following in scientific notation. Have to move the decimal 3 places to the left, so that the 1) 4700 new number has a value between 1 and 10. Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3. 4700 = 4.7 103 2) 0.00047 Have to move the decimal 4 places to the right, so that the new number has a value between 1 and 10. Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4. 0.00047 = 4.7 10-4 Martin-Gay, Developmental Mathematics 24 Scientific Notation To Write a Scientific Notation Number in Standard Form • Move the decimal point the same number of spaces as the exponent on 10. • If the exponent is positive, move the decimal point to the right. • If the exponent is negative, move the decimal point to the left. Martin-Gay, Developmental Mathematics 25 Scientific Notation Example Write each of the following in standard notation. 1) 5.2738 103 Since the exponent is a positive 3, we move the decimal 3 places to the right. 5.2738 103 = 5273.8 2) 6.45 10-5 Since the exponent is a negative 5, we move the decimal 5 places to the left. 00006.45 10-5 = 0.0000645 Martin-Gay, Developmental Mathematics 26 Operations with Scientific Notation Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Example Perform the following operations. 1) (7.3 10-2)(8.1 105) = (7.3 · 8.1) (10-2 · 105) = 59.13 103 = 59,130 2) 1 . 2 10 4 10 9 4 1 .2 4 10 4 10 9 0 . 3 10 5 0 . 000003 Martin-Gay, Developmental Mathematics 27 § 12.3 Introduction to Polynomials Polynomial Vocabulary Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator. Martin-Gay, Developmental Mathematics 29 Polynomial Vocabulary In the polynomial 7x5 + x2y2 – 4xy + 7 There are 4 terms: 7x5, x2y2, -4xy and 7. The coefficient of term 7x5 is 7, of term x2y2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term. Martin-Gay, Developmental Mathematics 30 Types of Polynomials Monomial is a polynomial with 1 term. Binomial is a polynomial with 2 terms. Trinomial is a polynomial with 3 terms. Martin-Gay, Developmental Mathematics 31 Degrees Degree of a term To find the degree, take the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a4b3c is 8 (remember that c can be written as c1). Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9x3 – 4x2 + 7 is 3. Martin-Gay, Developmental Mathematics 32 Evaluating Polynomials Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Example Find the value of 2x3 – 3x + 4 when x = 2. 2x3 – 3x + 4 = 2( 2)3 – 3( 2) + 4 = 2( 8) + 6 + 4 =6 Martin-Gay, Developmental Mathematics 33 Combining Like Terms Like terms are terms that contain exactly the same variables raised to exactly the same powers. Warning! Only like terms can be combined through addition and subtraction. Example Combine like terms to simplify. x2y + xy – y + 10x2y – 2y + xy = x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together) = (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y Martin-Gay, Developmental Mathematics 34 § 12.4 Adding and Subtracting Polynomials Adding and Subtracting Polynomials Adding Polynomials Combine all the like terms. Subtracting Polynomials Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms. Martin-Gay, Developmental Mathematics 36 Adding and Subtracting Polynomials Example Add or subtract each of the following, as indicated. 1) (3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3 = 4x2 + 3x – 3x – 8 + 3 = 4x2 – 5 2) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8 3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7) = – a2 + 1 – a2 + 3 + 5a2 – 6a + 7 = – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11 Martin-Gay, Developmental Mathematics 37 Adding and Subtracting Polynomials In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression. You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically. Martin-Gay, Developmental Mathematics 38 § 12.5 Multiplying Polynomials Multiplying Polynomials Multiplying polynomials • If all of the polynomials are monomials, use the associative and commutative properties. • If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms. Martin-Gay, Developmental Mathematics 40 Multiplying Polynomials Example Multiply each of the following. 1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3 2) (4x2)(3x2 – 2x + 5) = (4x2)(3x2) – (4x2)(2x) + (4x2)(5) = 12x4 – 8x3 + 20x2 (Distributive property) (Multiply the monomials) 3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5) = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20 Martin-Gay, Developmental Mathematics 41 Multiplying Polynomials Example Multiply (3x + 4)2 Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4). (3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4) = 9x2 + 12x + 12x + 16 = 9x2 + 24x + 16 Martin-Gay, Developmental Mathematics 42 Multiplying Polynomials Example Multiply (a + 2)(a3 – 3a2 + 7). (a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7) a4 – 3a3 + 7a + 2a3 – 6a2 + 14 = a4 – a3 – 6a2 + 7a + 14 = Martin-Gay, Developmental Mathematics 43 Multiplying Polynomials Example Multiply (3x – 7y)(7x + 2y) (3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y) = 21x2 + 6xy – 49xy + 14y2 = 21x2 – 43xy + 14y2 Martin-Gay, Developmental Mathematics 44 Multiplying Polynomials Example Multiply (5x – 2z)2 (5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z) = 25x2 – 10xz – 10xz + 4z2 = 25x2 – 20xz + 4z2 Martin-Gay, Developmental Mathematics 45 Multiplying Polynomials Example Multiply (2x2 + x – 1)(x2 + 3x + 4) (2x2 + x – 1)(x2 + 3x + 4) = (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4) = 2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4 = 2x4 + 7x3 + 10x2 + x – 4 Martin-Gay, Developmental Mathematics 46 Multiplying Polynomials You can also use a vertical format in arranging the polynomials to be multiplied. In this case, as each term of one polynomial is multiplied by a term of the other polynomial, the partial products are aligned so that like terms are together. This can make it easier to find and combine like terms. Martin-Gay, Developmental Mathematics 47 § 12.6 Special Products The FOIL Method When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms Martin-Gay, Developmental Mathematics 49 Using the FOIL Method Example Multiply (y – 12)(y + 4) (y – 12)(y + 4) Product of First terms is y2 (y – 12)(y + 4) Product of Outside terms is 4y (y – 12)(y + 4) Product of Inside terms is -12y (y – 12)(y + 4) Product of Last terms is -48 F O I L (y – 12)(y + 4) = y2 + 4y – 12y – 48 = y2 – 8y – 48 Martin-Gay, Developmental Mathematics 50 Using the FOIL Method Example Multiply (2x – 4)(7x + 5) L F F O I L (2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5) I O = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20 We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product. Martin-Gay, Developmental Mathematics 51 Special Products In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products. Squaring a Binomial (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 Multiplying the Sum and Difference of Two Terms (a + b)(a – b) = a2 – b2 Martin-Gay, Developmental Mathematics 52 Special Products Although you will arrive at the same results for the special products by using the techniques of this section or last section, memorizing these products can save you some time in multiplying polynomials. Martin-Gay, Developmental Mathematics 53 § 12.7 Dividing Polynomials Dividing Polynomials Dividing a polynomial by a monomial Divide each term of the polynomial separately by the monomial. Example 12 a 36 a 15 3 3a 12 a 3a 3 36 a 3a 4 a 12 2 15 3a 5 a Martin-Gay, Developmental Mathematics 55 Dividing Polynomials Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide. Martin-Gay, Developmental Mathematics 56 Dividing Polynomials 168 43 7256 43 29 5 258 37 6 344 32 Divide 43 into 72. Multiply 1 times 43. Subtract 43 from 72. Bring down 5. Divide 43 into 295. Multiply 6 times 43. Subtract 258 from 295. Bring down 6. Divide 43 into 376. Multiply 8 times 43. Subtract 344 from 376. Nothing to bring down. We then write our result as Martin-Gay, Developmental Mathematics 168 32 . 43 57 Dividing Polynomials As you can see from the previous example, there is a pattern in the long division technique. Divide Multiply Subtract Bring down Then repeat these steps until you can’t bring down or divide any longer. We will incorporate this same repeated technique with dividing polynomials. Martin-Gay, Developmental Mathematics 58 Dividing Polynomials 4x 5 7 x 3 28 x 23 x 15 2 28 x 12 x 2 35 x 15 35 x 15 Divide 7x into 28x2. Multiply 4x times 7x+3. Subtract 28x2 + 12x from 28x2 – 23x. Bring down – 15. Divide 7x into –35x. Multiply – 5 times 7x+3. Subtract –35x–15 from –35x–15. Nothing to bring down. So our answer is 4x – 5. Martin-Gay, Developmental Mathematics 59 Dividing Polynomials 2 x 10 2 2 x 7 4x 6 x 8 2 4 x 14 x 20x 8 20x 70 78 Divide 2x into 4x2. Multiply 2x times 2x+7. Subtract 4x2 + 14x from 4x2 – 6x. Bring down 8. Divide 2x into –20x. Multiply -10 times 2x+7. Subtract –20x–70 from –20x+8. Nothing to bring down. We write our final answer as 2 x 10 Martin-Gay, Developmental Mathematics 78 ( 2 x 7) 60