Raffles Institution Y4 WS1 Polynomials and Identities

Page 1 of 19 RAFFLES INSTITUTION RAFFLES PROGRAMME 2023 YEAR 4 MATHEMATICS TOPIC 2: REMAINDER & FACTOR THEOREMS AND PARTIAL FRACTIONS (MATH 1) WORKSHEET 1 Name: ( ) Class: 4 ( ) Date: WORKSHEET 1: POLYNOMIALS AND IDENTITIES think! Add Math Textbook A Chapter 4 p.54 Blended Learning Online (SDL) Students to access HeyMath! Lesson Year 4 / Remainder and Factor Theorem / Polynomial Identities / Introduction [5:20] (1) INTRODUCTION TO POLYNOMIAL The table shows some examples of polynomials and non-polynomials in one variable. Polynomials Non-polynomials 2 43xx−+ 11x x+− 54322746 1x x x xx+ − + −+ 1 23xx −−+ 4 52 2xx− 3xx− 8 3 24 12 xxx−− KEY UNDERSTANDING(S) Students will understand that  A polynomial can be written as LEARNER OUTCOMES At the end of this worksheet, students will be able to  Define a polynomial  Write a polynomial in the form  Distinguish between identities and equations.  Use long division to divide polynomials

Page 2 of 19 Compare the powers of x in polynomials with those in non-polynomials. What do you notice? Why is the number ‘8’ a polynomial? (2) DEFINITIONS A polynomial in x is an algebraic expression consisting of terms with non -negative integral powers of x only. NOTE: 1. 12 2 1 2 2 10, , , ... , , , nn n nn nax a x a x ax ax a−− −− are known as the terms of the polynomial. 1 2 210, , , ... , , , nn naa a aaa−− are called the coefficients of the polynomial. 0a is called the constant term. 2. The degree of a polynomial in x is the highest power of x that occurs in the polynomial. 3. Consider the polynomial 325 7 46xxx+ −+ . 35x is called the leading term as it contains the highest power of x . The coefficient of the leading term is called the leading coefficient. By convention, we usually arrange the terms of a polynomial in descending powers of the variable. 4. Commonly used notations for polynomials in x include P( )x , Q( )x , f( )x and g( )x . 5. P( )a is the value of P( )x when xa= . For example, to find the value of the polynomial 32P( ) 5 7 4 6xxxx= + −+ when 3x= , we substitute 3x= into P( )x : 32P(3) 5(3) 7(3) 4(3) 6 192 = + −+ = The general form of a polynomial in is , where the index is a non -negative integer and the coefficients are real numbers.

Page 3 of 19 6. Polynomials can be classified according to their degree and the number of terms. Polynomial Degree Name by degree No. of terms Name by no. of terms 8 7 11x−− 2 43xx−+ 3212 5xxx− + −+ 4 52 2xx− 54322746 1x x x xx+ − + −+ EG 1 Which of the following are polynomials? (a) 1x x+ (b) ( ) 3 21 9x−− (c) ( ) 2 5ex − (d) 12xx+− (e) 422 53 1x y xy y x− + +− (f) 2lg 3xx+ EG 2 Given that 42P( ) 2 3 7x xx x= −+− is a polynomial in x , determine the following: (a) 4Coefficient of x = _____ (b) 2Coefficient of x = _____ (c) Degree of P( )x = _____ (d) P(2) = _________________________ (e) P( 1)−= ________________________

Page 4 of 19 (3) IDENTITIES If two polynomials 32P( )x Ax Bx Cx D= + ++ and 32Q( ) 2 3 4x x xx= + −+ are equivalent, then the equation P () Q ()xx= is true for all real values of x , i.e. 2A= , 3B= , 1C =− and 4D= . We call this equation an identity. We use the symbol ‘≡’ to denote an identity, i.e. P () Q ()xx≡ . EG 3 Find the values of A , B and C if ( )( ) ( ) 24 37 1 3 1x x Ax x Bx C+ −≡ − + + −+ Method 1: Compare coefficients Method 2: Substitute suitable values of x Why can we arbitrarily choose any value of x for substitution?

Page 5 of 19 Method 3: Combination of Method 1 and Method 2 Steps: 1. Compare coefficients of highest power terms 2. Substitute factor value(s) of x 3. Compare constant terms (or sub. 0x= ) 4. Substitute other value(s) of x or compare coefficients of other powers of x Computational Thinking - Opportunities for Algorithmic Thinking The method of finding unknown constants allow opportunities for algorithmic thinking: - Write out the general approach as a sequence of steps - Work out the steps to find unknown constants in identity EG 4 Given that ( )( ) 32 22 51x x x x Ax Bx C D+ ++≡ − + + + , find the values of A , B , C and D .

Page 6 of 19 EG 5 Given ( )( ) 324 6 1 2 1 Q( )x x x x x ax b− +≡ − + + + , where Q( )x is a polynomial, find the values of a and b . Blended Learning Online (Optional) Students to access HeyMath! Lesson Year 4 / Remainder and Factor Theorem / Polynomial Identities / Examples 1 to 3 [9:43] to consolidate learning before doing Homework 1. HOMEWORK 1 LEVEL 1 1. Given that ( )( )( ) 323 5 45 1 2x x x x x A Bx C+ − −≡ − + + + , find the values of A , B and C . [Ans: 2, 3, 1ABC= = =− ] 2. think! Add Math Textbook A p.60 Ex 4A Q6c Given that ( )( )( ) 322 1 1 2 6 28Ax x x B C x x x+ − − += + −+ for all values of x , find the values of A , B and C . [Ans: 1, 3, 14AB C== −= ] 3. think! Add Math Textbook A p.60 Ex 4A Q10 The expression ( )( ) ( ) 215ax b x c x+ −+ + is equal to 18 for all values of x . find the values of a , b and c . [Ans: 3, 3, 3abc= −= −= ]

Page 7 of 19 LEVEL 2 1. think! Add Math Textbook A p.60 Ex 4A Q6d Given that ( )( )( ) 432 22 13 19 5 1 4 1 1x x x x x Ax x Bx C− + + += − + + + + for all values of x , find the values of A , B and C . [Ans: 2, 3, 5AB C== −= ] 2. Given ( )( ) 325 6 2 1 2 Q( )x x x x x x px q+ −+= + − + + for all values of x , find the values of p and q . [Ans: 20, 24pq= = ]

Page 8 of 19 3. Given that ( )( ) 43 2 2 22 4 11 16 3 2 7x x Ax x Bx x x C Dx+ + − +≡ − +++ + , find the values of A , B , C and D . [Ans: 9, 2, 3, 5A BC D= −= = −= − ]

Page 9 of 19 (4) LONG DIVISION OF POLYNOMIALS Let us recall long division of positive integers, e.g. divide 451 by 6: 7 5 6 4 5 1 ( ) 4 2 3 1 ( ) 3 0 1 − − We can express the dividend 451 in terms of the divisor, quotient and remainder as follows: 451 6 75 1= ×+ Dividend = Divisor ×Quotient + Remainder In general, if a polynomial is divided by another polynomial , then and .

Page 10 of 19 EG 6 By long division, find the quotient and remainder for each of the following divisions. Hence, express the polynomial in the form Dividend Divisor Quotient Remainder≡× + . (a) ( ) ( ) 32 74 3x x xx+ − ÷+ (b) ( ) ( ) 324 4 67 21xxx x+ −+÷ −