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Lesson6 - LESSON 6 LONG DIVISION AND SYNTHETIC DIVISION...

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LESSON 6 LONG DIVISION AND SYNTHETIC DIVISION Example Find ) 3 4 ( ) 20 16 12 14 3 2 ( 2 2 3 4 5 + - ÷ - + + - - x x x x x x x . 3 5 2 11 4 9 12 3 20 16 3 15 20 5 12 20 5 6 8 2 20 16 12 14 3 2 3 4 2 3 2 2 2 3 4 2 3 4 3 4 5 2 3 4 5 2 - + - - + - - + - + - + - + - - + + - - + - x x x x x x x x x x x x x x x x x x x x x x x NOTE: 3 4 5 2 3 6 8 2 ) 3 4 ( 2 x x x x x x + - = + - 2 3 4 2 2 15 20 5 ) 3 4 ( 5 x x x x x x + - = + - 9 12 3 ) 3 4 ( 3 2 2 - + - = + - - x x x x The expression 3 4 2 + - x x is called the divisor in the division. The function 3 4 ) ( 2 + - = x x x b is called the divisor function. The expression 20 16 12 14 3 2 2 3 4 5 - + + - - x x x x x is called the dividend in the division. The function 20 16 12 14 3 2 ) ( 2 3 4 5 - + + - - = x x x x x x a is called the dividend function. The expression 3 5 2 2 3 - + x x is called the quotient in the division. The function 3 5 2 ) ( 2 3 - + = x x x q is called the quotient function. The expression 11 4 - x is called the remainder in the division. The function 11 4 ) ( - = x x r is called the remainder function. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1320
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We have that 3 4 11 4 3 5 2 3 4 20 16 12 14 3 2 2 2 3 2 2 3 4 5 + - - + - + = + - - + + - - x x x x x x x x x x x x Multiplying both sides of this equation by 3 4 2 + - x x , we have that 20 16 12 14 3 2 2 3 4 5 - + + - - x x x x x = ) 11 4 ( ) 3 5 2 ( ) 3 4 ( 2 3 2 - + - + + - x x x x x Let a and b be polynomials. Then ) ( ) ( ) ( ) ( ) ( x b x r x q x b x a + = . The degree of the remainder polynomial r is less than the degree of divisor polynomial b , written deg r < deg b . Multiplying both sides of the equation ) ( ) ( ) ( ) ( ) ( x b x r x q x b x a + = by ) ( x r , we have that ) ( ) ( ) ( ) ( x r x q x b x a + = . Example Find ) 2 3 ( ) 7 21 17 12 ( 2 3 4 - ÷ + - - x x x x . 6 9 3 4 5 12 18 7 18 18 27 0 27 6 9 21 9 8 12 7 0 21 17 12 2 3 2 3 2 2 2 3 2 3 3 4 2 3 4 - - - - + - + - + - + - + - - - - + + - - - x x x x x x x x x x x x x x x x x x x x Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1320
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NOTE: 3 4 3 8 12 ) 2 3 ( 4 x x x x - = - 2 3 2 6 9 ) 2 3 ( 3 x x x x + - = - - x x x x 18 27 ) 2 3 ( 9 2 + - = - - 12 18 ) 2 3 ( 6 + - = - - x x The quotient function is 6 9 3 4 ) ( 2 3 - - - = x x x x q and the remainder function is 5 ) ( - = x r . We have that ) 5 ( ) 6 9 3 4 ( ) 2 3 ( 7 21 17 12 2 3 2 3 4 - + - - - - = + - - x x x x x x x .
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Lesson6 - LESSON 6 LONG DIVISION AND SYNTHETIC DIVISION...

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