Explanation Procedure used: Procedure for Synthetic Division: “1. Write the coefficients of f ( x ) . Be certain that the powers are in descending order and that zeros are inserted for missing powers. 2. Carry down the left coefficient, then multiply it by r , and place this product under the second coefficient of the top line. 3. Add the two numbers in the second column and place the result below. Multiply this sum by r and place the product under the third coefficient of the top line. 4. Continue this process until the bottom row has as many numbers as the top row”. Complex Conjugate Root Theorem: “If the coefficients of the equation f ( x ) = 0 and real and a + b j ( b ≠ 0 ) is a complex root, then its conjugate, a − b j , is also a root”. Calculation: It is given that r 1 = 3 j is the root of the equation s 4 − 8 s 3 − 72 s − 81 = 0 . By complex conjugate root theorem, r 2 = − 3 j also root of the equation s 4 − 8 s 3 − 72 s − 81 = 0 . The powers of x are in descending order, write down the coefficient of f ( s ) = s 4 − 8 s 3 − 72 s − 81 and zeros are inserted for missing powers and then place 3 j to the right end. 1 − 8 0 − 72 − 81 3 j Bring down the left coefficient 6 to the bottom line and multiply it by 3 j , and place the product value 3 j under the second coefficient –3 of the top line. Proceed in this similar manner, to find the quotient and remainder. 1 − 8 0 − 72 − 81 3 j 3 j − 9 − 24 j 72 − 27 j 81 _ 1 − 8 + 3 j − 9 − 24 j − 27 j 0 − 3 j − 3 j 24 j 27 j _ 1 − 8 − 9 0 The numbers presented in the last row denotes the coefficients of the quotient and the remainder

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Chapter 15.2, Problem 20E

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The remaining roots of the equation s4−8s3−72s−81=0 using synthetic division.

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Chapter 15.2, Problem 19E

Chapter 15.2, Problem 21E

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The remaining roots of the equation s 4 − 8 s 3 − 72 s − 81 = 0 using synthetic division.

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