Definition: Polynomial of real variable x of order n, with integer coefficients, is of the form: p(x) = a 0 + a 1 x + a 2 x² +...

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1 Worksheet : Polynomial Name: Date: Coverage:. Definition and properties of polynomial. Algebraic operation: addition, subtraction, multiplication and division of polynomials.. Remainder theorem Definition: Polynomial of real variable of order n, with integer coefficients, is of the form: p() = a 0 a a ²... a n n where : a 0, a, a,..., a n are integers and a n 0. n is a non-negative integer, known as the order of the polynomial. Note : When a n =, the p() is called monik. Eample:. p() = is a monik of order 5.. q() = - 7 ² ½ is a monik of order.. r() = 5-5 is not a monik since a n. Value of a polynomial The value of a polynomial p() at = a is p(a), obtained by substituting with a. Eample: ) Let p() = ² Then: p() =..² = 6 6 = 0. Using similar procedure, we obtain: p(0) = 0 p() = 7 p( ) = and so on. ) Let q() = ² Then: q(0) = =, q() = 0, q(-) = 0, and so on.

2 Algebraic operation of polynomials Let: p() = a 0 a a ²... a m m, and q() = b 0 b b ²... b n n. Addition of two polynomials m > n m = n m < n p()q() = (a 0 b 0 ) (a b ) (a b ) ²... (a n b n ) n a n n... a m m p()q() = (a 0 b 0 ) (a b ) (a b ) ²... (a n b n ) n p()q() = (a 0 b 0 ) (a b ) (a b ) ²... (a m b m ) m b m m...b n n For simplicity: add coefficients of the same power of. Eample: Let p() = ² and q() = ² Re-write: p() = ² = 0 0 q() = ² = Then: p() q() = (0 0)( - - 0) = (0 ) ( )² (0) (0) = ². Subtraction of two polynomials m > n m = n m < n p()-q() = (a 0 -b 0 ) (a -b ) (a -b ) ²... (a n -b n ) n a n n... a m m p()-q() = (a 0 -b 0 ) (a -b ) (a -b ) ²... (a n -b n ) n p()-q() = (a 0 -b 0 ) (a -b ) (a -b ) ²... (a m -b m ) m - b m m - - b n n For simplicity: subtract coefficients of the same power of. Eample: Let p() = ² and q() = ²

3 Re-write: p() = ² = 0 0 q() = ² = Then: p() - q() = (0 0) - ( - - 0) Give a try! q()-p() = = (0 - -) (- -)² ( - 0) (0 - ) = ² -. Multiplication of two polynomials Multiply all combination of coefficients of the two polynomials, then group the result based on the power of. Let: p() = a 0 a a ²... a m m, and q() = b 0 b b ²... b n n Then, p() q() = a 0 b 0 (a 0 b a b 0 )... a m b n mn Where the coefficient of k is : a 0 b k a b k a b k... a k b 0 Hard to imagine? Lets look at the eamples. Eample:. Let p() = 5² and q() = ² Then: p() q() = ( 5²) ( ² ) = ( ² ) 5² ( ² ) = ( 6 ) (5² ) = 5² Let p() = 5² and q() = ² Then: p() q() = ( 5²) ( ² ) =( ² ) ( ² ) 5² ( ² ) = ( ² ) ( 6 ) (5² ) = ²

4 Eercise Determine order of the polynomial, and determine whether it is a monic or not! Polynomial Order Monic? A p() = NO B k(t) = t t² 7 C q() = ² D l(t) = t 5 t 0 E r() = 00 F m(t) = t² t G f(s) = ( s) H n(z) = z² z 0 I g(s) = s s² J o(z) = (z z²)² Using polynomials in number, determine the following! p() q() p() - q() k(t) l(t) f(s)q(s) k(t) l(t) n(z) o(z) f(s) - q(s) n(z) - o(z) k(s) q(s) m() r()

5 Using polynomials in number, calculate the following: p() m() - n( ) k( ) f() n( ) q(0) n(-) ( o(-) k(-) ) l( ) g(-) o(5) r(0) m(-) ( p() - l()) q( ) Determine the result of the following: a. ( ) ( ²) b. (5 ) - ( ) c. ( ) ( ²) d. ( ) ( ) e. ( ²)

6 . Division of two polynomial Definition: Let: f() = a 0 a a ²... a m m, and g() = b 0 b b ²... b n n 0. Then: There wll be h() and r() such that f() = h() g() r(); Where: r() = 0,or order of r() < order of g(). h() = result of division of f() by g() r() = remainder term of division. How to divide?. Standard division, as in real numbers division Eample : Let: f() = ² and g() = f( ) Then :? g ( ) = Procedure: 0 Steps:. Divide st term of f() by st term of g(): 0 : =. Multiply by ² - ², 9 Put under f(). Subtract - ² from f(). Repeat step to, until order of the remainder < order of the divisor. So, the result is h() = ² with the remainder r() =. Synthetic division or Horner Method 6

7 Eample : Let: f() = ² and g() = f( ) Then :? g ( ) = Procedure: Step: write the coefficient, in order, from the highest order to the lowest. f() = ² 0 Step : if the divisor is of the form g() = c then it is written as = c = Step : 0 9 (*) (**) Sum the th column = This is the remainder; r()= Sum the st column =, multiply by, put the result in the nd column, below - Sum the rd column =, multiply by, put the result in the th column, below Sum the nd column =, multiply by, put the result in the rd column, below 0 (*) is the coefficients of the result: h() = ² (**) is the remainder: r() =. 7

8 Properties of polynomials division Remainder Theorem : Division of f() by ( c) f( ) Let f() be the polynomial. If ( ) c = h with the remainder r(), then r()=f(c ). Can be written as: f() = ( c) h() f (c), Remainder Theorem : Division of f() by (a b) h If f() is divided by a b, then the result is and the remainder is b h ( ) b f f( ) = ( a b) f a a a where f ( ) h ( ) = b a Eample Let f() = ² and g() = 5. f ( ) Find using methods, and verify the result using the above theorem! g ( ) Answer: ( ) a Method Hence: the result is h() = ² 7, and the remainder is: r() = 6

9 Method. Horner (*) (**) Hence, the result is: h() = ² 7 and the remainder is r() = 6. Verify: using Remainder theorem r() = f(5) = 5.5².5 = = 6 Let f() = ² 5 and g() =. f ( ) Find using methods, and verify the result using the above theorem! g ( ) Answer: Method. 5 Hence, the result is: h() = and the remainder is r() =. Method. Horner Coefficients of f() : 5 Divisor: = ; a = dan b = 5 9

10 (*) (**) Using remainder theorem, the result is: ( ) () * = = = a a a h, and the remainder is: r() = (**) = = f Using remainder theorem, we obtain the same remainder, that is: r() = f(/) = (/) 5.(/) = 9/ 5/ = Let f() = ² and g() =. Find the result,h(), and the remainder, r(), of ( ) ( ) f g using methods! Answer: Method. 0 7 Hence: the result is ) ( = h,and

11 the remainder is r ( ) Method. Horner Coefficients of f(): = 7 f() = ² 0 - Since the divisor is g() = a b = b/a. g() = = ½ Therefore: / 0 / / -/ h The result is: a The remainder is / -/ 7/ ( ) () (*) (**) * = = = a 7 f =,and

12 Eercise. Using standard division method, determine the result and the remainder of division f() by g(), where the polynomials are given as the following: a f() =, g() = b f() =, g() = ² c f() =, g() = d f() = ², g() = ². Using Horner s method, determine the result and the remainder of division f() by g(), where the polynomials are given as the following: a. f() = 5, g() = d. f() = 5, g() = b. f() = 9² 7, g() = e. f() = 5² 6, g() = c. f() = ² 6, g() = f. f() = 5, g() =

13 . Verify the result of no. using remainder theorem. a. d. b. e. c. f.. Using either standard or Horner s method, determine the result and the remainder of division f() by g(), where the polynomials are given as the following: a.f () = ² 6 5, g() = d. f() = 6² 9, g() = b. f() = 9² 6 0, g() = e. f() = ² 0, g() = 5 c. f() = 6² 5, g() = f. f() = ², g() = 5. Verify the result of no. using remainder theorem. a. d. b. e. c. f.

14 6. Solve the following: a Find a if f() = a ² 6 is divisible by g() = b Find b if f() = ² b is divisible by g() = c Find c and the result, h(), if f() = 6 ² 9 c is divisible by g() =

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