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Polynomials

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

  2. Introduction What are Polynomials? A polynomial in x consists of a finite number of terms of the form axnwhere 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,...})

  3. The following are examples of polynomials: 2x4 A one term polynomial is called a monomial. -5x6 + 7.9x2 A two term polynomial is called a binomial. 2x2 + 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. 19y1/2 + 5 is not a polynomial because the exponent on the variable is not a whole number.

  4. The following are examples of polynomials in more than one variable: 2x4 y2 is a monomial in x and y -5x6yz + 7.9x2yz2 is a binomial in x, y & z 2x2w + 3xw2 - w3 is a trinomial in x and w

  5. How many terms does the following polynomial have? 7x4 - 3x2 + 4x3 - 9x +5 The polynomial 7x4 - 3x2 + 4x3 - 9x +5 has 5 terms. 1 2 3 4 5

  6. 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: - 3x2 + 4x3 - 9x + x4 +5 would be written x4 + 4x3 - 3x2 - 9x + 5

  7. Degree of a Term The degree of a term is the sum of the exponents on all variables. For example: the degree of 5x2y3z is (2 + 3 + 1) or 6 For the polynomial x4 + 4x3 - 3x2 - 9x + 5 the degree of each term from left to right is 4, 3, 2, 1, and 0. The constant 5 is equal to 5x0, thus it has degree 0.

  8. Degree of a Polynomial The degree of a polynomial is the largest degree of any one term. Thus in the preceding polynomial, x4 + 4x3 - 3x2 - 9x + 5, the degree would be 4. What is the degree of x7 + 4x8 - 3x9 - 9x3 + 5 ?

  9. The degree of x7 + 4x8 - 3x9 - 9x3 + 5 is 9 the highest degreed term.

  10. What are Like Terms? Like terms are terms with the same variables raised to the same powers. For example: 5x2y3 is like -4y3 x2 but is not like 5y3x x is like .35x but is not like x2 Which of the following pairs are pairs of like terms? (A) 3xy and 2yz (B) -2xyz and 5xyz (C) 3x2 and 4 x3

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

  12. Adding Polynomials 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, -5x2 has numerical coefficient -5, and x has numerical coefficient 1]

  13. 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? ]

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

  15. Example 2: (5x2 + 8x - 7) + (-9x3 - 8x2 - 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(5x2 + 8x - 7) + 1(-9x3 - 8x2 - 7x + 3)] 5x2 + 8x - 7 - 9x3 - 8x2 - 7x + 3 Combining like terms -9x3 - 3x2 + x - 4

  16. 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 (-10x4 + 8x2 - 1) and ( 2x4 - 5x2 + 4x + 3) Write -10x4 + 8x2 - 1 2x4 - 5x2 + 3 + 4x like terms under each other -8x4 + 3x2 + 2 + 4x add columns

  17. 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.

  18. Example 2: (8x2 - 2x - 5) - (x3 - 9x2 - 2x + 5) • 1(8x2 - 2x - 5) - 1(x3 - 9x2 - 2x + 5) • Remove grouping symbols • 8x2 - 2x - 5 - x3 + 9x2 + 2x - 5 • Add like terms • - x3 + 17x2 -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 5x3 - 3x -10 from 8x3 - 2x • 8x3 - 2x 8x3 - 2x -5x3 + 3x + 10 becomes -1(5x3 - 3x -10) 3x3 + x + 10

  19. Multiplication of Polynomials Multiplying a Monomial by a Monomial Example 1: (-2x6)(3x4) = -2 . 3 . x6 . x4 = -6x(6+4) = -6x10 Example 2: (10x2y)(3x9y2) Write the answer before you click your mouse. 10 . 3 . x2. x9. y1 . y2 = 30x(2+9)y(1+2) = 30x11y3

  20. Example 3: (-4x7y0)(-9x0yz3) = Write the answer before you click your mouse. -4 . -9 . x7. x0. y0. y1. z3 36x(7+0)y(0+1)z3 36x7yz3

  21. Multiplying a Monomial by a Polynomial Example 1: Distribute -2x thru parenthesis to each term -2x ( x2 - 3x + 9) = -2x(x2) - (-2x)3x + (-2x)9 = -2x3 + 6x2 - 18x Example 2: 3a2(-2a3 + 8a - 10) = Write your answer before you click your mouse. 3a2(-2a3) + 3a2(8a) + 3a2(-10) = -6a5 + 24a3 - 30a2 Example 3: (5x2 - 4x + 6) (3x) = Write your answer before you click your mouse. 5x2 (3x) - 4x (3x) + 6 (3x) = 15x3 - 12x2 +18x

  22. Example 4: (-3x4 - 5x2 + 1) (-3x2) Write your answer before you click your mouse. = -3x4 (-3x2) - 5x2 (-3x2) + 1 (-3x2) 9x6 + 15x4 - 3x2 Example 5: -2a( a3 + a2 - a + 4) = Write your answer before you click your mouse. -2a( a3) + -2a (a2 ) -2a( - a) + -2a(4) = -2a4 - 2a3 + 2a2 - 8a

  23. 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 2x2 First O stands for outside. (x + 4)(2x -5) The outside terms are x and -5 Outside More on the next slide.

  24. 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 Inside L stands for last. (x + 4)(2x -5) The last terms are 4 and -5 Their product is -20 Last More on the next slide.

  25. Putting all the products together we get: (x + 4)( 2x - 5) = 2x2 - 5x + 8x - 20 F O I L Combining like terms the final answer is 2x2 + 3x - 20 Example 1: Multiply (3y - 7)(5y - 6) First 3y(5y) = 15y2 Outside 3y(-6) = - 18y Inside -7 (5y) = - 35y Last -7 ( -6) = +42 Answer is 15y2 - 18y - 35y + 42 Final Answer is 15y2 - 53y + 42

  26. 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 + ad First Outside Then distribute b b(c + d) = bc + bd Inside Last Final Answer is ac + ad + bc + bd

  27. 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)( 4x2 - 3x -2) Because the second polynomial is not a binomial we cannot use FOIL. Instead multiply 2x by ( 4x2 - 3x -2) and then multiply 3 by ( 4x2 - 3x -2). The result is 2x( 4x2 - 3x - 2) = 8x3 - 6x2 - 4x then 3( 4x2 - 3x -2) = 12x2 - 9x -6 Now add down: 8x3 +6x2 - 13x -6

  28. 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.

  29. Division by a Monomial Divisor Example 1: Divide (3x4 - 5x3 +7x - 8) by 5x2 Write each term of the dividend as a fraction with a denominator of 5x2. Simplify each fraction to... 3x4 - 5x3 + 7x - 8 = 3x2 - x + 7 - 8 5x2 5x2 5x2 5x2 5 5x 5x2 Example 2: 9x3 - 4x2 + 8x - 6 3x Write 9x3 - 4x2 + 18x - 6 = 3x2 - 4x + 6 - 2 3x 3x 3x 3x 3 3x

  30. Division by a Polynomial with 2 or more terms. Divide (12 + X2 ) 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 + X2 must be written as X2 + 12. 2) If there are any missing exponents in your dividend , you make space for them by adding a zero term. Quotient Divisor Dividend X+3 X2 +0X + 12

  31. X+3 X2 + 5X + 12 X X+3 X2 + 5X + 12 X2 + 3X X X+3 X2 + 5X + 12 -X2 - 3X 2X + 12 Example 1: Divide (X2 + 5X + 12) by ( X + 3) Set up the long division problem. Divide the first term X2 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!

  32. X + 2 + 6 X+3 X X + 3 X+3 X2 + 5X + 12 -X2 - 3X 2X + 12 X2 + 5X + 12 -X2 - 3X 2X + 12 -2X - 6 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) + 2 - - 2X + 6 6 Write the final answer with the remainder in the form below. The remainder is 6. Good Job!

  33. X- 2 X2 + 0X - 5 X X- 2 X2 + 0X - 5 X2 - 2x X X- 2 X2 + 0X - 5 -X2 + 2X +2X - 5 Example 2: Divide (X2 - 5) by ( X - 2) Set up the long division problem. Divide the first term X2 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!

  34. X + 2 X- 2 X2 + 0X - 5 -X2 + 2X > 2X - 5 2X - 4 X + 2 + -1 x - 2 X - 2 X2 + 0X - 5 -X2 - 2X 2X - 5 -2X + 4 -1 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. - + -1 Write the final answer in the form on left.

  35. 2X + 1 8X3 + 0X2 + 0X - 1 4X2 2X + 1 4X2 2X+1 8X3 + 0X2 + 0X - 1 8X3 + 4X2 Example 3: Divide 8X3 - 1 2X + 1 Set up the long division problem before you click the mouse. Step 1: Divide 8X3 by 2X Write answer before you click. 8X3 + 0X2 + 0X - 1 Step2: Multiply 4X2 by divisor. Write answer before you click. We are not finished yet so continue onto the next page!

  36. 4X2- 2X 2X+1 8X3 + 0X2 + 0X - 1 -8X3 - 4X2 - 4X2 + 0X - 1 -4X2 - 2X 4X2 2X + 1 8X3 + 0X2 + 0X - 1 8X3 +4X2 Step 3: Subtract by changing signs and bring down left over terms. - - - 4X2 + 0X - 1 Repeat Steps 1 - 3 again Step 1: Divide - 4X2 by 2X Step2: Multiply -2X by divisor 2x + 1. + + Step3: Subtract

  37. + + + + 4x2 2x -4x2 - 2x 4X2- 2X 4X2- 2X 2X+1 2X+1 8X3 + 0X2 + 0X - 1 -8X3 - 4X2 - 4X2 + 0X - 1 8X3 + 0X2 + 0X - 1 -8X3 - 4X2 - 4X2 + 0X - 1 Step 3: Subtract by changing signs and bring down left over terms. 2x - 1 Repeat Steps 1 - 3 again Step 1: Divide 2X by 2X + 1 + -2 2x +1 Step2: Multiply 1 by divisor. Step 3: Subtract 2x -1 - Go To Practice Problems - 2x + 1 -2