A.4 Polynomial Division; Synthetic Division

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1 SECTION A.4 Polynomial Division; Synthetic Division 977 A.4 Polynomial Division; Synthetic Division OBJECTIVES 1 Divide Polynomials Using Long Division 2 Divide Polynomials Using Synthetic Division 1 Divide Polynomials Using Long Division The procedure for dividing two polynomials is similar to the procedure for dividing two integers. EXAMPLE 1 Dividing Two Integers Divide 842 by 15. Solution Divisor : ; Quotient ; Dividend ; 5 # 15 (Subtract) ; 6 # 15 (Subtract) ; Remainder So, = In the long division process detailed in Example 1, the number 15 is called the divisor, the number 842 is called the dividend, the number 56 is called the quotient, and the number 2 is called the remainder. To check the answer obtained in a division problem, multiply the quotient by the divisor and add the remainder. The answer should be the dividend. 1Quotient21Divisor2 + Remainder = Dividend For example, we can check the results obtained in Example 1 as follows: = = 842 To divide two polynomials, we first must write each polynomial in standard form. The process then follows a pattern similar to that of Example 1. The next example illustrates the procedure. EXAMPLE 2 Dividing Two Polynomials Find the quotient and the remainder when 3x 3 + 4x 2 + x + 7 is divided by x Solution Each polynomial is in standard form. The dividend is 3x 3 + 4x 2 + x + 7, and the divisor is x

2 978 APPENDIX Review STEP 1: Divide the leading term of the dividend, 3x 3, by the leading term of the divisor, x 2. Enter the result, 3x, over the term 3x 3, as follows: 3x x x 3 + 4x 2 + x + 7 STEP 2: Multiply 3x by x and enter the result below the dividend. 3x x x 3 + 4x 2 + x + 7 3x 3 + 3x ; 3x # (x 2 + 1) = 3x 3 + 3x q Notice that we align the 3x term under the x to make the next step easier. STEP 3: Subtract and bring down the remaining terms. 3x x x 3 + 4x 2 + x + 7 3x 3 + 3x 4x 2-2x + 7 ; Subtract (change the signs and add) ; Bring down the 4x 2 and the 7. STEP 4: Repeat Steps 1 3 using 4x 2-2x + 7 as the dividend. 3x + 4 x x 3 + 4x 2 + x + 7 3x 3 + 3x 4x 2-2x + 7 4x x + 3 ; ; Divide 4x 2 by x 2 to get 4. ; Multiply x by 4; subtract. Since does not divide -2x evenly (that is, the result is not a monomial), the process ends.the quotient is 3x + 4, and the remainder is -2x + 3. x 2 CHECK: (Quotient)(Divisor) + Remainder = 13x + 421x x + 32 = 3x 3 + 4x 2 + 3x x + 32 = 3x 3 + 4x 2 + x + 7 = Dividend Then 3x 3 + 4x 2 + x + 7 x = 3x x + 3 x The next example combines the steps involved in long division. EXAMPLE 3 Dividing Two Polynomials Find the quotient and the remainder when x 4-3x 3 + 2x - 5 is divided by x 2 - x + 1

3 SECTION A.4 Polynomial Division; Synthetic Division 979 Solution In setting up this division problem, it is necessary to leave a space for the missing x 2 term in the dividend. Divisor : Subtract : Subtract : Subtract : x 2-2x - 3 x 2 - x + 1x 4-3x 3 + 2x - 5 x 4 - x 3 + x 2-2x 3 - x 2 + 2x - 5-2x 3 + 2x 2-2x -3x 2 + 4x - 5-3x 2 + 3x - 3 x - 2 ; Quotient ; Dividend ; Remainder CHECK: (Quotient)(Divisor) + Remainder = 1x 2-2x - 321x 2 - x x - 2 = x 4 - x 3 + x 2-2x 3 + 2x 2-2x - 3x 2 + 3x x - 2 = x 4-3x 3 + 2x - 5 = Dividend As a result, x 4-3x 3 + 2x - 5 x 2 = x 2 x - 2-2x x + 1 x 2 - x + 1 The process of dividing two polynomials leads to the following result: Theorem Let Q be a polynomial of positive degree and let P be a polynomial whose degree is greater than the degree of Q. The remainder after dividing P by Q is either the zero polynomial or a polynomial whose degree is less than the degree of the divisor Q. 2 NOW WORK PROBLEM 9. Divide Polynomials Using Synthetic Division To find the quotient as well as the remainder when a polynomial of degree 1 or higher is divided by x - c, a shortened version of long division, called synthetic division, makes the task simpler. To see how synthetic division works, we will use long division to divide the polynomial 2x 3 - x by x x 2 + 5x + 15 x - 32x 3 - x x 3-6x 2 # 5x x 2-15x 15x x CHECK: (Divisor) (Quotient) + Remainder ; Quotient ; Remainder = 1x x 2 + 5x = 2x 3 + 5x x - 6x 2-15x = 2x 3 - x 2 + 3

4 980 APPENDIX Review The process of synthetic division arises from rewriting long division in a more compact form, using simpler notation. For example, in the long division on p. 979, the terms in blue are not really necessary because they are identical to the terms directly above them. With these terms removed, we have 2x 2 + 5x + 15 x - 32x 3 - x x 2 5x 2-15x 15x Most of the x s that appear in this process can also be removed, provided that we are careful about positioning each coefficient. In this regard, we will need to use 0 as the coefficient of x in the dividend, because that power of x is missing. Now we have 2x 2 + 5x + 15 x We can make this display more compact by moving the lines up until the numbers in color align horizontally. 2x 2 + 5x + 15 x ~ Row 4 Because the leading coefficient of the divisor is always 1, we know that the leading coefficient of the dividend will also be the leading coefficient of the quotient. So we place the leading coefficient of the quotient, 2, in the circled position. Now, the first three numbers in row 4 are precisely the coefficients of the quotient, and the last number in row 4 is the remainder. Thus, row 1 is not really needed, so we can compress the process to three rows, where the bottom row contains both the coefficients of the quotient and the remainder. x (subtract) Recall that the entries in row 3 are obtained by subtracting the entries in row 2 from those in row 1. Rather than subtracting the entries in row 2, we can

5 SECTION A.4 Polynomial Division; Synthetic Division 981 change the sign of each entry and add.with this modification, our display will look like this: x (add) Notice that the entries in row 2 are three times the prior entries in row 3. Our last modification to the display replaces the x - 3 by 3. The entries in row 3 give the quotient and the remainder, as shown next (add) Quotient Remainder 2x 2 5x Let s go through an example step by step. EXAMPLE 4 Using Synthetic Division to Find the Quotient and Remainder Use synthetic division to find the quotient and remainder when x 3-4x 2-5 is divided by x - 3 Solution STEP 1: Write the dividend in descending powers of x. Then copy the coefficients, remembering to insert a 0 for any missing powers of x STEP 2: Insert the usual division symbol. In synthetic division, the divisor is of the form x - c, and c is the number placed to the left of the division symbol. Here, since the divisor is x - 3, we insert 3 to the left of the division symbol STEP 3: Bring the 1 down two rows, and enter it in row p 1 STEP 4: Multiply the latest entry in row 3 by 3, and place the result in row 2, one column over to the right STEP 5: Add the entry in row 2 to the entry above it in row 1, and enter the sum in row

6 982 APPENDIX Review STEP 6: Repeat Steps 4 and 5 until no more entries are available in row STEP 7: The final entry in row 3, the -14, is the remainder; the other entries in row 3, the 1, -1, and -3, are the coefficients (in descending order) of a polynomial whose degree is 1 less than that of the dividend. This is the quotient. Thus, Quotient = x 2 - x - 3 Remainder = -14 CHECK: (Divisor)(Quotient) + Remainder = 1x - 321x 2 - x = 1x 3 - x 2-3x - 3x 2 + 3x = x 3-4x 2-5 = Dividend Let s do an example in which all seven steps are combined. EXAMPLE 5 Using Synthetic Division to Verify a Factor Use synthetic division to show that x + 3 is a factor of 2x 5 + 5x 4-2x 3 + 2x 2-2x + 3 Solution The divisor is x + 3 = x , so we place -3 to the left of the division symbol. Then the row 3 entries will be multiplied by -3, entered in row 2, and added to row Because the remainder is 0, we have 1Divisor21Quotient2 + Remainder = 1x x 4 - x 3 + x 2 - x + 12 = 2x 5 + 5x 4-2x 3 + 2x 2-2x + 3 As we see, x + 3 is a factor of 2x 5 + 5x 4-2x 3 + 2x 2-2x + 3. As Example 5 illustrates, the remainder after division gives information about whether the divisor is, or is not, a factor. NOW WORK PROBLEMS 23 AND 33.

7 982 APPENDIX Review A.4 Assess Your Understanding Concepts and Vocabulary 1. To check division, the expression that is being divided, the dividend, should equal the product of the and the plus the. 2. To divide 2x 3-5x + 1 by x + 3 using synthetic division, the first step is to write. 3. True or False: In using synthetic division, the divisor is always a polynomial of degree 1, whose leading coefficient is 1.

8 SECTION A.4 Polynomial Division; Synthetic Division True or False: means x 3 + 3x 2 + 2x + 1 x + 2 = 5x 2-7x x + 2. Skill Building In Problems 5 20, find the quotient and the remainder. Check your work by verifying that 1Quotient21Divisor2 + Remainder = Dividend 5. 4x 3-3x 2 + x + 1 divided by x x 3-3x 2 + x + 1 divided by x x 4-3x 2 + x + 1 divided by x x 5-3x 2 + x + 1 divided by 2x x 4-3x 3 + x + 1 divided by 2x 2 + x x 3 + x 2-4 divided by x x 2 + x 4 divided by x 2 + x x 3 - x 2 + x - 2 divided by x + 2 3x 3 - x 2 + x - 2 divided by x 2 5x 4 - x 2 + x - 2 divided by x x 5 - x 2 + x - 2 divided by 3x 3-1 3x 4 - x 3 + x - 2 divided by 3x 2 + x + 1-3x 4-2x - 1 divided by x x 2 + x 4 divided by x 2 - x x 3 - a 3 divided by x - a 20. x 5 - a 5 divided by x - a In Problems 21 32, use synthetic division to find the quotient and remainder. 21. x 3 - x 2 + 2x + 4 divided by x x 3 + 2x 2-3x + 1 divided by x x 3 + 2x 2 - x + 3 divided by x x 5-4x 3 + x divided by x x 6-3x 4 + x divided by x x x divided by x x 5-1 divided by x x 3 + 2x 2 - x + 1 divided by x + 2 x 4 + x divided by x - 2 x 5 + 5x 3-10 divided by x x divided by x x divided by x + 1 In Problems 33 42, use synthetic division to determine whether x - c is a factor of the given polynomial x 3-3x 2-8x + 4; x x 3 + 5x 2 + 8; x x 4-6x 3-5x + 10; x x x ; x x 6-64x 4 + x 2-15; x x 4-15x 2-4; x - 2 2x 6-18x 4 + x 2-9; x + 3 x 6-16x 4 + x 2-16; x x 4 - x 3 + 2x - 1; x x 4 + x 3-3x + 1; x Find the sum of a, b, c, and d if Discussion and Writing x 3-2x 2 + 3x + 5 x + 2 = ax 2 + bx + c + d x When dividing a polynomial by x - c, do you prefer to use long division or synthetic division? Does the value of c make a difference to you in choosing? Give reasons.

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