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Polynomials By Dr. Julia Arnold Tidewater Community College Copyright 10/19/2002

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Introduction What are Polynomials? A polynomial in x consists of a finite number of terms of the form ax n where a can be any Real number but n must be a whole number. (Recall a term is any algebraic expression separated from another algebraic expression by “+” or “-” signs. Whole numbers are {0,1,2,3,4,...})

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The following are examples of polynomials: 2x 4 A one term polynomial is called a monomial. -5x 6 + 7.9x 2 A two term polynomial is called a binomial. 2x 2 + 3x - 1 A three term polynomial is called a trinomial. The following are not polynomials: 3x -2 - 4x -1 + 2 is not a polynomial because the exponents on the variables are not whole numbers. 19y 1/2 + 5 is not a polynomial because the exponent on the variable is not a whole number.

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The following are examples of polynomials in more than one variable: 2x 4 y 2 is a monomial in x and y -5x 6 yz + 7.9x 2 yz 2 is a binomial in x, y & z 2x 2 w + 3xw 2 - w 3 is a trinomial in x and w

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The polynomial 7x 4 - 3x 2 + 4x 3 - 9x +5 has 5 terms. 1 2 3 4 5 How many terms does the following polynomial have? 7x 4 - 3x 2 + 4x 3 - 9x +5

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Descending Powers Writing a polynomial in descending powers means to begin with the term having the largest exponent on the variable and then proceeding to the lowest. For example: - 3x 2 + 4x 3 - 9x + x 4 +5 would be written x 4 + 4x 3 - 3x 2 - 9x + 5

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Degree of a Term The degree of a term is the sum of the exponents on all variables. For example: the degree of 5x 2 y 3 z is (2 + 3 + 1) or 6 For the polynomial x 4 + 4x 3 - 3x 2 - 9x + 5 the degree of each term from left to right is 4, 3, 2, 1, and 0. The constant 5 is equal to 5x 0, thus it has degree 0.

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Degree of a Polynomial The degree of a polynomial is the largest degree of any one term. Thus in the preceding polynomial, x 4 + 4x 3 - 3x 2 - 9x + 5, the degree would be 4. What is the degree of x 7 + 4x 8 - 3x 9 - 9x 3 + 5 ?

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The degree of x 7 + 4x 8 - 3x 9 - 9x 3 + 5 is 9 the highest degreed term.

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What are Like Terms? Like terms are terms with the same variables raised to the same powers. For example: 5x 2 y 3 is like -4y 3 x 2 but is not like 5y 3 x x is like.35x but is not like x 2 Which of the following pairs are pairs of like terms? (A) 3xy and 2yz (B) -2xyz and 5xyz (C) 3x 2 and 4 x 3

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The answer is (B) -2xyz is like 5xyz because both have the same variables raised to the same powers or exponents.

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To add polynomials (1) remove the grouping symbols, (2) find the like terms of the polynomial, and then (3) add the numerical coefficients of the like terms. [Note: the numerical coefficient is the number with the variables; i.e. 3xyz has numerical coefficient 3, -5x 2 has numerical coefficient -5, and x has numerical coefficient 1] Adding Polynomials

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Example 1: (2x +3) + (5x - 6) First remove grouping symbols 2x + 3 + 5x - 6 Next find the like terms 2x + 5x + 3 - 6 Add numerical coefficients (2+5)x + (3 - 6) = 7x - 3 [Question: Does 2x + 5x = (2 + 5)x remind you of a property stressed earlier in the course? ]

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The distributive property: a(b + c) = ab + ac

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Example 2: (5x 2 + 8x - 7) + (-9x 3 - 8x 2 - 7x + 3) [Remember to remove the grouping symbols, multiply by whatever number is in front of the grouping symbols, using the distributive property. That number in this problem is 1 for both polynomials. 1(5x 2 + 8x - 7) + 1(-9x 3 - 8x 2 - 7x + 3)] 5x 2 + 8x - 7 - 9x 3 - 8x 2 - 7x + 3 Combining like terms -9x 3 - 3x 2 + x - 4

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Adding in Columns If you prefer, you can use the column method of adding polynomials. Like terms are placed under each other. Example 3: Add (-10x 4 + 8x 2 - 1) and ( 2x 4 - 5x 2 + 4x + 3) Write -10x 4 + 8x 2 - 1 2x 4 - 5x 2 + 3 + 4x like terms under each other -8x 4 + 3x 2 + 2 + 4x add columns

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Subtracting Polynomials To subtract polynomials you must remove the grouping symbols by multiplying the first expression by 1 and the second expression by -1. Example 1: (2x +3) - (5x - 6) 1(2x + 3) -1 ( 5x - 6) First remove grouping symbols 2x + 3 - 5x + 6 (Note: Multiplying by -1 causes the signs to change in your expression.) 2x - 5x and 3 + 6 Add the like terms -3x + 9 is the result.

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Example 2: (8x 2 - 2x - 5) - (x 3 - 9x 2 - 2x + 5) 1(8x 2 - 2x - 5) - 1(x 3 - 9x 2 - 2x + 5) Remove grouping symbols 8x 2 - 2x - 5 - x 3 + 9x 2 + 2x - 5 Add like terms - x 3 + 17x 2 -10 (the answer) Subtracting in Columns Be very careful when using this method. You must make sure you change all the signs of the polynomial being subtracted. Example 3: Subtract 5x 3 - 3x -10 from 8x 3 - 2x 8x 3 - 2x 8x 3 - 2x -5x 3 + 3x + 10 3x 3 + x + 10 -1(5x 3 - 3x -10) becomes

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Multiplication of Polynomials Multiplying a Monomial by a Monomial Example 1: (-2x 6 )(3x 4 ) = -2. 3. x 6. x 4 = -6x (6+4) = -6x 10 Example 2: (10x 2 y)(3x 9 y 2 ) Write the answer before you click your mouse. 10. 3. x 2. x 9. y 1. y 2 = 30x (2+9) y (1+2) = 30x 11 y 3

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Example 3: (-4x 7 y 0 )(-9x 0 yz 3 ) = -4. -9. x 7. x 0. y 0. y 1. z 3 36x (7+0) y (0+1) z 3 36x 7 yz 3 Write the answer before you click your mouse.

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Multiplying a Monomial by a Polynomial Example 1: Distribute -2x thru parenthesis to each term -2x ( x 2 - 3x + 9) = -2x(x 2 ) - (-2x)3x + (-2x)9 = -2x 3 + 6x 2 - 18x Example 2: 3a 2 (-2a 3 + 8a - 10) = 3a 2 (-2a 3 ) + 3a 2 (8a) + 3a 2 (-10) = -6a 5 + 24a 3 - 30a 2 Write your answer before you click your mouse. Example 3: (5x 2 - 4x + 6) (3x) = 5x 2 (3x) - 4x (3x) + 6 (3x) = 15x 3 - 12x 2 +18x Write your answer before you click your mouse.

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Example 4: (-3x 4 - 5x 2 + 1) (-3x 2 ) = -3x 4 (-3x 2 ) - 5x 2 (-3x 2 ) + 1 (-3x 2 ) 9x 6 + 15x 4 - 3x 2 Example 5: -2a( a 3 + a 2 - a + 4) = -2a( a 3 ) + -2a (a 2 ) -2a( - a) + -2a(4) = -2a 4 - 2a 3 + 2a 2 - 8a Write your answer before you click your mouse.

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Multiplying a Binomial by a Binomial To multiply two binomials together we use an acronym called FOIL to help us remember the products. F stands for first. In the problem (x + 4)(2x -5) The first terms are x and 2x Their product is 2x 2 O stands for outside. (x + 4)(2x -5) The outside terms are x and -5 More on the next slide.

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Multiplying a Binomial by a Binomial Continued FOIL stands for First, Outside, Inside, Last. I stands for inside. (x + 4)(2x -5) The inside terms are 4 and 2x Their product is 8x L stands for last. (x + 4)(2x -5) The last terms are 4 and -5 Their product is -20 More on the next slide.

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Putting all the products together we get: (x + 4)( 2x - 5) = 2x 2 - 5x + 8x - 20 F O I L Combining like terms the final answer is 2x 2 + 3x - 20 Example 1: Multiply (3y - 7)(5y - 6) First 3y(5y) = 15y 2 Outside 3y(-6) = - 18y Inside -7 (5y) = - 35y Last -7 ( -6) = +42 Answer is 15y 2 - 18y - 35y + 42 Final Answer is 15y 2 - 53y + 42

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Do you see that each term of the first polynomial is multiplied by each term in the second polynomial? Example 2: Multiply (a + b)(c + d) Distribute a thru (c + d) a(c + d) = ac Final Answer is ac + ad + bc + bd First + ad Outside Then distribute b b(c + d) =bc+ bd InsideLast

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If you understand this basic premise: that each term of the first polynomial is multiplied by each term in the second polynomial, then it will be an easy transition to multiply polynomials containing more than two terms. Example 3: Multiply (2x + 3)( 4x 2 - 3x -2) Because the second polynomial is not a binomial we cannot use FOIL. Instead multiply 2x by ( 4x 2 - 3x -2) and then multiply 3 by ( 4x 2 - 3x -2). The result is 2x( 4x 2 - 3x - 2) = 8x 3 - 6x 2 - 4x then 3( 4x 2 - 3x -2) = 12x 2 - 9x -6 Now add down: 8x 3 +6x 2 - 13x -6

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Division of Polynomials There are two types of division techniques. The first kind that will be illustrated is division by a monomial. The second kind is for division by any other type of polynomial. Occasionally, monomial division produces some unexpected answers. If you try to use the “second method” for dividing by a monomial, you may find yourself unable to complete the task.

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Division by a Monomial Divisor Example 1: Divide (3x 4 - 5x 3 +7x - 8) by 5x 2 Write each term of the dividend as a fraction with a denominator of 5x 2. Simplify each fraction to... 3x 4 - 5x 3 + 7x - 8 = 3x 2 - x + 7 - 8 5x 2 5x 2 5x 2 5x 2 5 5x 5x 2 Example 2: 9x 3 - 4x 2 + 8x - 6 3x Write 9x 3 - 4x 2 + 18x - 6 = 3x 2 - 4x + 6 - 2 3x 3x 3x 3x 3 3x

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Divisor Dividend X+3X 2 +0X + 12 Divide (12 + X 2 ) by (X + 3) In a long division problem you must follow two set-up rules. 1) The dividend must be arranged in descending powers. Thus 12 + X 2 must be written as X 2 + 12. 2) If there are any missing exponents in your dividend, you make space for them by adding a zero term. Division by a Polynomial with 2 or more terms. Quotient

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X+3 X 2 + 5X + 12 Divide the first term X 2 by the first term in the divisor, X. Write the result above 5X. Multiply X by the divisor X + 3 and write the answer below the dividend matching like terms as you go. Subtract the bottom line by changing the signs of the bottom line you just wrote. When finished bring down the next term, which is 12. We are not finished yet so continue onto the next slide! Example 1: Divide (X 2 + 5X + 12) by ( X + 3) Set up the long division problem. X 2 + 5X + 12 X 2 + 3X X+3 X X 2 + 5X + 12 -X 2 - 3X 2X + 12 X+3 X

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2X + 6 Divide the first term 2X by the first term X. The answer is 2. Write 2 above the 12. (click mouse) Multiply 2(X + 3) = 2X + 6 Write answer below 2X + 12. (click mouse) Subtract by changing signs. (Click mouse twice) X 2 + 5X + 12 -X 2 - 3X 2X + 12 -2X - 6 6 X + 3 X + 2 + 6 X+3 Write the final answer with the remainder in the form below. X 2 + 5X + 12 -X 2 - 3X 2X + 12 X+3 X + 2 - - 6 The remainder is 6.

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X- 2 X 2 + 0X - 5 Divide the first term X 2 by the first term in the divisor, X. Write the result above 0X. Multiply X by the divisor X - 2 and write the answer below the dividend matching like terms as you go. Subtract the bottom line by changing the signs of the bottom line you just wrote. When finished bring down the next term, which is -5 We are not finished yet so continue onto the next page! Example 2: Divide (X 2 - 5) by ( X - 2) Set up the long division problem. X 2 + 0X - 5 X 2 - 2x X- 2 X X 2 + 0X - 5 -X 2 + 2X +2X - 5 X- 2 X

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Divide the first term 2X by the first term X. The answer is 2. Write +2 above the -5. Multiply 2(X - 2) = 2X - 4 Write answer below 2X - 5. Subtract by changing signs. The remainder is -1. Write the final answer in the form on left. X 2 + 0X - 5 -X 2 + 2X > 2X - 5 2X - 4 X- 2 X + 2 X 2 + 0X - 5 -X 2 - 2X 2X - 5 -2X + 4 X - 2 X + 2 + -1 x - 2 - +

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8X 3 + 0X 2 + 0X - 1 2X + 1 4X 2 We are not finished yet so continue onto the next page! Example 3: Divide 8X 3 - 1 2X + 1 Set up the long division problem before you click the mouse. Step 1: Divide 8X 3 by 2X Write answer before you click. Step2: Multiply 4X 2 by divisor. Write answer before you click. 8X 3 + 0X 2 + 0X - 1 8X 3 + 4X 2 2X+1 4X 2 8X 3 + 0X 2 + 0X - 1 2X + 1

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8X 3 + 0X 2 + 0X - 1 8X 3 + 4X 2 Repeat Steps 1 - 3 again Step 1: Divide - 4X 2 by 2X Step2: Multiply -2X by divisor 2x + 1. 8X 3 + 0X 2 + 0X - 1 -8X 3 - 4X 2 - 4X 2 + 0X - 1 2X+1 4X 2 - 2X Step3: Subtract 4X 2 Step 3: Subtract by changing signs and bring down left over terms. -- -4X 2 - 2X ++ - 4X 2 + 0X - 1

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Repeat Steps 1 - 3 again Step 1: Divide 2X by 2X Step2: Multiply 1 by divisor. Step 3: Subtract by changing signs and bring down left over terms. Go To Practice Problems ++ 8X 3 + 0X 2 + 0X - 1 -8X 3 - 4X 2 - 4X 2 + 0X - 1 2X+1 4X 2 - 2X -4x 2 - 2x ++ 8X 3 + 0X 2 + 0X - 1 -8X 3 - 4X 2 - 4X 2 + 0X - 1 2X+1 4X 2 - 2X 4x 2 2x 2x -1 + 1 2x + 1 - - -2 + -2 2x +1 2x - 1 Step 3: Subtract

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