Victoria School AM #4 Polynomials and Partial Fractions 2024

Secondary 4 Additional Mathematics Polynomials and Partial Fractions 1 Teacher’s Name: Class Reg No. Date: 4 Chapter 4: Polynomials and Partial Fractions Reference Book: Additional Maths 360 Textbook, Marshall Cavendish 4.1 Polynomials and Identities You will learn how to, • Add, subtract, multiply and divide polynomials, • Find the unknown constants in a polynomial identity, • Divide one polynomial by another using the division algorithm. What are Polynomials? “Polynomial” is a Greek word which literally means ‘many terms’. (a) A polynomial is a sum of terms, each of the form nax , where a is a constant and the power n is a non-negative integer. Examples of polynomials include x + 1, 3x2 – 4x + 5, x3 – 1 4 x2 + 1 2 , and – x3. (b) In the term nax , a is called the coefficient of nx . (c) The degree or order of a polynomial in x is the highest power of x. Example 1 Determine whether each of the following is a polynomial. Give a brief reason for your answer. If it is a polynomial, state its degree. Polynomial Degree/ Order x + 1 1 3x2 – 4x + 5 2 x3 – 1 4 x2 + 1 2 3 – x3 3 4 0 x2 + 1 2 x 2 (x – 2)(x + 1) 2 322 ++ xx Not a polynomial as it has a term with a fractional power. (d) A polynomial is often denoted by P(x), Q(x), f(x) etc. If Q(x) = 322 ++ xx , then the value of this polynomial at x = 2 is denoted by Q(2) = 22 + 2(2) + 3 = 11. → More generally, the value of the polynomial, Q(x) at x = a, is denoted by Q(a).

Secondary 4 Additional Mathematics Polynomials and Partial Fractions 2 Example 2 Evaluate P(3) and P(−1) if P(x) = x3 – 2x + 1. ( ) ( ) ( ) ( ) ( ) ( ) 3 3 P 3 3 2 3 1 = 22 P 1 1 2 1 1 = 4 = − + − = − − − + Addition and Subtraction and Multiplication of Polynomials When two polynomials are added, subtracted or multiplied, the result is still a polynomial. Addition and subtracting polynomials can be done by combining like terms while multiplication of two polynomials can be done through either using the distributive law. Example 3 Consider the polynomials 2( ) 1P x x x= + + and ( ) 22 3 2.Q x x x= − + Find (i) ( ) ( ),Q x P x− (ii) ( ) ( )2.P x Q x+ ( ) ( ) ( ) 22 22 2 (i) 2 3 2 1 2 3 2 1 41 − = − + − + + = − + − − − = − + Q x P x x x x x x x x x xx ( ) ( ) ( ) 22 22 2 (ii) 2 1 2 2 3 2 1 4 6 4 5 5 5 + = + + + − + = + + + − + = − + P x Q x x x x x x x x x xx Example 4 By observation, find the coefficient of x3 and x2 in the expansion of (i) ( )( ) 323 – 2 5 –1 2 1 ,++x x x x (ii) ( )( ) 222 – 3 1 .+xx ( ) ( ) ( ) ( ) 3 2 (i) Coefficient of 3 1 2 2 1 Coefficient of 2 1 5 2 8 =− =− =− + = x x ( ) ( ) 3 2 (ii) Coefficient of 0 Coefficient of 2 1 3 1 1 = =− =− x x

Secondary 4 Additional Mathematics Polynomials and Partial Fractions 3 Finding Unknowns in Identities Consider the polynomials x + 3 and x2 – x. Now, x + 3 = x2 – x holds only for