# Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Exponents and Polynomials.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Exponents and Polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.1 Exponents

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = 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, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate each of the following expressions. 3434 = 3 3 3 3= 81 (–5) 2 = (– 5)(–5)= 25 –6 2 = – (6)(6) = –36 (2 4) 3 = (2 4)(2 4)(2 4)= 8 8 8= 512 3 4 2 = 3 4 4= 48 Evaluating Exponential Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate each of the following expressions. Evaluating Exponential Expressions Example a.) Find 3x 2 when x = 5. b.) Find –2x 2 when x = –1. 3x 2 = 3(5) 2 = 3(5 · 5)= 3 · 25 –2x 2 = –2(–1) 2 = –2(–1)(–1)= –2(1) = 75 = –2

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If m and n are positive integers and a is a real number, then a m · a n = a m+n 3 2 · 3 4 = 3 6 x 4 · x 5 = x 4+5 z 3 · z 2 · z 5 = z 3+2+5 (3y 2 )(– 4y 4 )= 3 · y 2 (– 4) · y 4 = 3(– 4)(y 2 · y 4 ) = – 12y 6 = 3 2+4 = x 9 = z 10 The Product Rule For example,

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Don’t forget that In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifying the exponential expression. 3 5 ∙ 3 7 = 9 12 3 5 ∙ 3 7 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = 3 12 12 factors of 3, not 9. Add exponents. Common base not kept. 5 factors of 3. 7 factors of 3.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Don’t forget that if no exponent is written, it is assumed to be 1.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If m and n are positive integers and a is a real number, then (a m ) n = a mn For example, (2 3 ) 3 = 2 9 (x4)2(x4)2 = x 8 = 2 3·3 = x 4·2 The Power Rule

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If n is a positive integer and a and b are real numbers, then (ab) n = a n · b n The Power of a Product Rule For example, = 5 3 · (x 2 ) 3 · y 3 = 125x 6 y 3 (5x 2 y) 3

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If n is a positive integer and a and c are real numbers, then The Power of a Quotient Rule For example,

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The Quotient Rule For example, If m and n are positive integers and a is a real number, then Group common bases together.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. a 0 = 1, a ≠ 0 Note: 0 0 is undefined. For example, 5050 = 1 (xyz 3 ) 0 = x 0 · y 0 · (z 3 ) 0 = 1 · 1 · 1 = 1 –x0–x0 = –(x 0 ) = – 1 Zero Exponent

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.2 Negative Exponents and Scientific Notation

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Negative Exponents Using the quotient rule from section 3.1, But what does x -2 mean?

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 is a real number other than 0, and n is an integer, then Negative Exponents

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplifying Expressions Simplify. Write each result using positive exponents only. Example Don’t forget that since there are no parentheses, x is the base for the exponent –4. Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 18 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify. Write each result using positive exponents only. Simplifying Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 19 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify by writing each of the following expressions with positive exponents. 1) 2) (Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from the numerator to the denominator, or vice versa, and switch the exponent to its opposite value.) Simplifying Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 20 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If m and n are integers and a and b are real numbers, then: Product Rule for exponents a m · a n = a m+n Power Rule for exponents (a m ) n = a mn Power of a Product (ab) n = a n · b n Power of a Quotient Quotient Rule for exponents Zero exponent a 0 = 1, a ≠ 0 Negative exponent Summary of Exponent Rules

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 21 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify by writing the following expression with positive exponents or calculating. Power of a quotient rulePower of a product rule Power rule for exponents Quotient rule for exponents Negative exponents Simplifying Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 the product of a number a, where 1 ≤ a < 10, and an integer power r of 10: a 10 r Scientific Notation

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 23 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Write a Number in Scientific Notation Step 1:Move the decimal point in the original number so that the new number has a value between 1 and 10 Step 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. Step 3:Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2. Scientific Notation

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 24 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Write each of the following in scientific notation. 4700 1) Move the decimal 3 places to the left, so that the 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 10 3 0.00047 2) 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 Scientific Notation Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 25 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In general, 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. Scientific Notation

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 26 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Write each of the following in standard notation. 5.2738 10 3 1) Since the exponent is a positive 3, we move the decimal 3 places to the right. 5.2738 10 3 = 5273.8 6.45 10 -5 2) Since the exponent is a negative 5, we move the decimal 5 places to the left. 00006.45 10 -5 = 0.0000645 Scientific Notation Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 27 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Operations with Scientific Notation Example Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Perform the following operations. = (7.3 · 8.1) (10 -2 · 10 5 ) = 59.13 10 3 = 59,130 (7.3 10 -2 )(8.1 10 5 ) 1) 2)

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.3 Introduction to Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 29 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Defining Term and Coefficient 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 finite sum of terms of the form ax n, where a is a real number and n is a whole number.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 30 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In the polynomial 7x 5 + x 2 y 2 – 4xy + 7 There are 4 terms: 7x 5, x 2 y 2, – 4xy and 7. The coefficient of term 7x 5 is 7, of term x 2 y 2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term. Defining Term and Coefficient

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 31 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Monomial is a polynomial with exactly one term. Binomial is a polynomial with exactly two terms. Trinomial is a polynomial with exactly three terms. Types of Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 32 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The degree of a term in one variable is the exponent on the variable. The degree of a constant is 0. The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of 9x 3 – 4x 2 + 7 is 3. The degree of a term with more than one variable is the sum of the exponents on the variables. The degree of the term 5a 4 b 3 c is 8 (remember that c can be written as c 1 ). Degrees

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Find the value of 2x 3 – 3x + 4 when x = – 2. = 2( – 2) 3 – 3( – 2) + 42x 3 – 3x + 4 = 2( – 8) + 6 + 4 = – 6 Evaluating Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 34 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Like terms are terms that contain exactly the same variables raised to exactly the same powers. Combine like terms to simplify. x 2 y + xy – y + 10x 2 y – 2y + xy Only like terms can be combined through addition and subtraction. Warning! = 11x 2 y + 2xy – 3y = (1 + 10)x 2 y + (1 + 1)xy + (– 1 – 2)y = x 2 y + 10x 2 y + xy + xy – y – 2y Group like terms. Combining Like Terms Example Use the distributive property. Simplify.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 36 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Add Polynomials Combine all the like terms. To Subtract Polynomials Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms. Adding and Subtracting Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 37 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Add: (3x – 8) + (4x 2 – 3x +3) = 4x 2 + 3x – 3x – 8 + 3 = 4x 2 – 5 = 3x – 8 + 4x 2 – 3x + 3 Example Adding and Subtracting Polynomials (3x – 8) + (4x 2 – 3x +3)

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 38 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. = 3a 2 – 6a + 11 Subtract. 1) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8 2) (– a 2 + 1) – (a 2 – 3) + (5a 2 – 6a + 7) = – a 2 + 1 – a 2 + 3 + 5a 2 – 6a + 7 = – a 2 – a 2 + 5a 2 – 6a + 1 + 3 + 7 Example Adding and Subtracting Polynomials 4 – (– y – 4) (– a 2 + 1) – (a 2 – 3) + (5a 2 – 6a + 7)

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 39 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Don’t forget to change the sign of each term in the polynomial being subtracted.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 40 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically. Adding and Subtracting Polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.5 Multiplying Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 42 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. Multiplying Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 43 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Multiply Two polynomials Multiply each term of the first polynomial by each term of the second polynomial, and then combine like terms. Multiplying Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply each of the following. 1) (3x 2 )( – 2x) = (3)( – 2)(x 2 · x) = – 6x 3 2) (4x 2 )(3x 2 – 2x + 5) = (4x 2 )(3x 2 ) – (4x 2 )(2x) + (4x 2 )(5) Apply the distributive property. = 12x 4 – 8x 3 + 20x 2 Multiply the monomials. 3) (2x – 4)(7x + 5)= 2x(7x + 5) – 4(7x + 5) = 14x 2 + 10x – 28x – 20 = 14x 2 – 18x – 20 Multiplying Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 45 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (3x + 4) 2. Remember that a 2 = a · a, so (3x + 4) 2 = (3x + 4)(3x + 4). (3x + 4) 2 = (3x + 4)(3x + 4)= (3x)(3x + 4) + 4(3x + 4) = 9x 2 + 12x + 12x + 16 = 9x 2 + 24x + 16 Multiplying Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 46 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (a + 2)(a 3 – 3a 2 + 7). (a + 2)(a 3 – 3a 2 + 7) = a(a 3 – 3a 2 + 7) + 2(a 3 – 3a 2 + 7) = a 4 – 3a 3 + 7a + 2a 3 – 6a 2 + 14 = a 4 – a 3 – 6a 2 + 7a + 14 Multiplying Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 47 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (3x – 7y)(7x + 2y). (3x – 7y)(7x + 2y)= (3x)(7x + 2y) – 7y(7x + 2y) = 21x 2 + 6xy – 49xy + 14y 2 = 21x 2 – 43xy + 14y 2 Multiplying Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 48 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (5x – 2z) 2. (5x – 2z) 2 = (5x – 2z)(5x – 2z)= (5x)(5x – 2z) – 2z(5x – 2z) = 25x 2 – 10xz – 10xz + 4z 2 = 25x 2 – 20xz + 4z 2 Multiplying Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 49 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (2x 2 + x – 1)(x 2 + 3x + 4) (2x 2 + x – 1)(x 2 + 3x + 4) = (2x 2 )(x 2 + 3x + 4) + x(x 2 + 3x + 4) – 1(x 2 + 3x + 4) = 2x 4 + 6x 3 + 8x 2 + x 3 + 3x 2 + 4x – x 2 – 3x – 4 = 2x 4 + 7x 3 + 10x 2 + x – 4 Multiplying Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 50 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Another convenient method for multiplying polynomials is to multiply vertically, similar to the way we multiply real numbers. 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. Multiplying Polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.6 Special Products

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 52 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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, Prealgebra & Introductory Algebra, 3ed 53 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. = y 2 – 8y – 48 Multiply (y – 12)(y + 4) (y – 12)(y + 4) Product of First terms is y 2 Product of Outside terms is 4y Product of Inside terms is – 12y Product of Last terms is – 48 (y – 12)(y + 4) = y 2 + 4y – 12y – 48 F O I L Using the FOIL Method Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 54 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (2x – 4)(7x + 5). (2x – 4)(7x + 5) = = 14x 2 + 10x – 28x – 20 F 2x(7x) F + 2x(5) O – 4(7x) I – 4(5) L O I L = 14x 2 – 18x – 20 We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product. Using the FOIL Method Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products. Special Products

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 56 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Squaring a Binomial A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 Special Products

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 57 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Squaring a Binomial Example: Multiply. (x + 6) 2 F OI L (x + 6) 2 = x 2 + 6x + 6x + 36 = x 2 + 12x + 36 The inner and outer products are the same. = (x + 6)(x + 6)

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 58 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Squaring a Binomial Example: Multiply. (12a – 3) 2 (12a – 3) 2 = 144a 2 – 72a + 9 = (12a) 2 – 2(12a)(3) + (3) 2 Example: Multiply. (x + y) 2 (x + y) 2 = x 2 + 2xy + y 2

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 59 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (a + b)(a – b) = a 2 – b 2 Special Products

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 60 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Sum and Difference of Two Squares Example: Multiply. (2x + 4)(2x – 4) = (2x)(2x) + (2x)(– 4) + (4)(2x) + (4)(– 4) F OI L (2x + 4)(2x – 4) = 4x 2 + (– 8x) + 8x + (–16) = 4x 2 – 16 The inner and outer products cancel.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 61 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Sum and Difference of Two Squares Example: Multiply. (5a + 3)(5a – 3) (5a + 3)(5a – 3) = 25a 2 – 9 = (5a) 2 – 3 2 Example: Multiply. (8c + 2d)(8c – 2d) (8c + 2d)(8c – 2d) = 64c 2 – 4d 2 = (8c) 2 – (2d) 2

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 62 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint When multiplying two binomials, you may always use the FOIL order or method. When multiplying any two polynomials, you may always use the distributive property to find the product.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 63 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. When multiplying two binomials, you may always use the FOIL order or method. When multiplying any two polynomials, you may always use the distributive property to find the product. Helpful Hint

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.7 Dividing Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 65 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing Polynomials To Divide a Polynomial by a Monomial Divide each term of the polynomial separately by the monomial.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing Polynomials Example Divide:

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 67 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To divide a polynomial by a polynomial other than a monomial, we use a process known as long division. Polynomial long division is similar to number long division, which we review on the next slide. Dividing Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 68 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 Dividing Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 69 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. Dividing Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 70 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 35  x Divide 7x into 28x 2. Multiply 4x times 7x+3. Subtract 28x 2 + 12x from 28x 2 – 23x. Bring down – 15. Divide 7x into –35x. Multiply – 5 times 7x+3. Subtract –35x–15 from –35x–15. Nothing to bring down. 15  So our answer is 4x – 5. Dividing Polynomials

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 71 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing Polynomials Notice that the division process is continued until the degree of the remainder polynomial is less than the degree of the divisor polynomial.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 72 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 86472 2  xxx x2 x x 14 4 2  20  x 10  7020  x 78 Divide 2x into 4x 2. Multiply 2x times 2x+7. Subtract 4x 2 + 14x from 4x 2 – 6x. Bring down 8. Divide 2x into –20x. Multiply -10 times 2x+7. Subtract –20x–70 from –20x+8. Nothing to bring down. 8   )72( 78  x x2 10  We write our final answer as Dividing Polynomials.