An Insight into Division Algorithm, Remainder and Factor Theorem


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1 An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the quotient to be a positive integer and allow it to be as large as possible, but that the remainder must be positive. The restriction results in the remainder being smaller than the divisor, otherwise the division is not complete, we should increase the quotient. Division in Polynomials A polynomial is an algebraic epression of the form n n a n + an a + a, where a, a,... a n are real numbers and n is a positive integer. When a polynomial P( ) is divided by another polynomial D, ( ) we stop the division when the remainder has degree less than the divisor D ( ) So we see at once that division is needed only if degree of D ( ) is not more than that of P( ). At any step of the division, we try to clear the term with the highest degree left from the dividend One case of division is of great interest to us, it is when the divisor is of degree, that is when D ( ) is of the form a b, then the remainder is a constant, that is, a term not involving The result is usually written in the simple form b P( ) = ( α) Q( ) + R, where the divisor appears as α = instead of a b a Q ( ) This can be written also as P( ) = ( a b)[ ] + R. (We must have already implied a that a when we say the divisor is a b ) We see that the remainder is the same whether we take the divisor as a b or α as long as b a = α Looking at P( ) = ( α) Q( ) + R again, we see that the LHS and RHS are actually identical epression, in fact the equality holds for any value of We may write P( ) ( α) Q( ) + R, meaning that both side are equal for all values of The result P( α ) = R follows This result states that P( α ) can be found by division or by substitution. If P( α ) =, we find a root α of the equation P ( ) = and at the same time a factor α of P ( )
2 Our interest is generally the solving of the equation P ( ) =, of degree or more The algorithm is: Work out P( α ), P( α), P( α),... until we get P( α ) = Use division or some other method to obtain P( ) = ( α) Q( ) Repeat now with Q ( ) When we get Q( β ) =, we shall get Q ( ) = ( β ) Q ( ) So, P( ) = ( α)( β ) Q ( ) The above process looks tedious. In practice, we will take α, α, α,... to be the easy choices such as,,, ,. then the rational numbers made out from factors of a and an if all coefficients of P( ) are integers After successfully putting P( ) = ( α) Q( ), try for the other roots, which are now in Q ( ) =, save repeating the failure cases as we try for P ( ) =, but do include the successful cases again. For enhanced effectiveness, leave out writing n n n,,,..., in the work for long division, but ensuring that the coefficients going to the correct positions. We do the long division, to see that the remainder ( the value of P( )) is zero. If so, we have found, at the same time, the linear factor and its corresponding quotient. Eample Solve the cubic equation = Solution Let f ( ) = We find that f () =, f () =, f ( ) = = So, + is a factor, or say = is a root By long division, f( ) = ( + )( 8+ ) Let q ( ) = 8+ By inspection, q ( ) = ( )( 6) We have f( ) = ( + )( )( 6) = =,, 6
3 Eercise (without solution attached). Factorize the epression Given + is a factor of 4 a b , find the values of a and b.. Find the remainder when 4. Find the remainder when 5. Factorize the polynomial 6. Factorize the polynomial + is divided by is divided by as far as possible 6a + a 6a as far as possible 7. Show that a is a factor of + ( a ) + ( a ) a. Find the other factor + 8. Show by Remainder Theorem that is a factor of + 9. Show that is a factor of 6. What is the other factor?. A polynomial leaves remainder when it is divided by or coefficient of is, what is the polynomial? +. If the
4 Eercise (with solution attached) 4. Factorize the epression 6+ 8 ( )( )( + + 4) 4. Given + is a factor of + a + b + 8, find the values of a and b. Solution 4 + = ( )( ), so and are factors of + a + b + 8 a =, b= 6. Find the remainder when 4. Find the remainder when 8 5. Factorize the polynomial ( + )( ) ( + ) + is divided by is divided by as far as possible 6. Factorize the polynomial 6a + a 6a as far as possible Try = a, = a, = a ( a)( a)( a) 7. Show that a is a factor of ( a)( + ) ( a ) ( a ) a + +. Find the other factor + 8. Show by Remainder Theorem that is a factor of + By substituting =, we get remainder zero + 9. Show that is a factor of 6. What is the other factor? By substituting =, we get remainder zero By division, we get the other factor + ( )( + ) 4
5 . A polynomial leaves remainder when it is divided by or +. If the coefficient of is, what is the polynomial? Let the polynomial be f ( ), then f ( )  is divisible by and + So f ( )  = ( )( + ), f ( ) = ( )( + ) + = + + 5
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