EP-Program - Strisuksa School - Roi-et Math : Polynomial Functions Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 00 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Polynomial Functions The function P ( ) below is an eample of polynomial: P 4 ( ).5 8 In general, polynomial functions of,such as or, multiplied by constants and added together. Eample of functions which are not polynomials are:,, 4,cos, sin tan, etc. The order or the degree of a polynomial is the highest power of.so the order of the polynomial P ( ) above is 4. If the variable in P ( ) is replaced by the number, the result in written P( ). In the eample above: 4 P().5 8 7 Algebra of polynomials Polynomial can be added, subtracted and multiplied by using ordinary algebra methods. The result can be simplified by the collection of terms with same power of. ( ) ( 4 0.5) 6.5 ( ) ( ) 7 When one polynomial is divided by another, long division is used. To divide by ( ) : 7 ( ) So the quotient when 9 7 7 is divide by is 7, and the remainder is.
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page. Remainder and factor theorems The remainder theorem states: The remainder when P ( ) is divide by is P( ). Proof So the remainder when P(). is divided by is P (). This confirms the result above. A particular case of the Remainder is the Factor Theorem: Proof divides P ( ) if and only if P ( ) 0 When trying to find factors of P, ( ) the first values of to try are the divisors of the constant term in P( ). If P( ) 0 then is a factor of P. ( ).. Eamples 5. Find the remainder when is divide by. Solution Use the Remainder Theorem above, putting. 5 P( ) ( ) ( ) ( ) The remainder is 4. a b is divisible by( )( ). Find the values of a andb. Solution Use the Factor Theorem twice, putting and : a b 0
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page a b : 8 8 a b 0 a b 6 Solve these simultaneously to obtain: a 5 and b 6. Factorise 4 6. Solution Use the Factor Theorem, trying first the divisor of 6. For, P( ) 4.For, P( ) 0 So we know that ( ) is a factor. Use the division process to find the other factor. 4 6 ( )( 5 6) This quadratic factor factorises to ( )( ). Hence: 4 6 ( )( )( ).. Eercises. Which of the following are polynomials? Write down the order of those which are (a) (b) 5 7 5 6 ( )( 7 ). Evaluate the following polynomial sums and products: (a) ( ) ( 4 4) (b) ( ) ( ) 5 ( ) ( 5) 4 ( ) ( ). Find the quotient and remainder in each of following division problems. (a) 4 (b) 5
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page 4 4 4 4 4. Find the remainder in each of the following divisions. (a) 5 (b) 7 4 5. The remainder when is divide by is5.find the value of a. a 6. The remainder when 5 is divide by is8.find the value of a a 7. When a b is divide by the remainder is, and when it is divide by the remainder is.find a andb.
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page 5 8. When b is divided by the remainder is, and when it is divided by the remainder is. Find a andb. 9. divide a. Find a. 0. Find k, given that is a factor of k.. is divisible by both and. Find a and b. a b. a b 5is divisible by ( )( ).Find a andb.. 4 divides a b 0.Find a and b, and find the third factor. 4. Factorise the following: (a) (b) 4 5 6 7 6
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page 6 (e) 9 7 6 (f) 4 4 4 (g) (h) 5 8 4 5. Solve the following equations. (a) 4 6 0 (b) 4 6 0 5 0 6 6 0
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page 7 Solution (to eercise..). b,, d,4.. (a) (b) (a) 4 5 4 7 5 6 5,97 (b) 4,8 4. (a) 8 (b) 9 7 8 5.5 4 8, 7 4.5,4.5 6. 5 9 7. a, b 6 8. a, b 7 7 9. 0.. a 0, b 4. a, b 4 4 4. a 5, b 4.( 5) 4. (a) ( )( )( ) (b) ( )( )( ) ( )( )( ) ( )( )( ) (e) ( )( )( ) (f) (g) ( )( )( ) ( ) ( )
EP.Program Strisuksa School Roi-et. Mathematics. Polynomials page 8 5. (h) ( )( ) (a),, (b),,,,,, =========================================================== References: Solomon, R.C. (997), A Level: Mathematics (4 th Edition), Great Britain, Hillman Printers(Frome) Ltd. More: (in Thai) http://home.kku.ac.th/wattou/service/m/4.pdf