Revision on Polynomials

Do you remember what a monomial is? A monomial is an algebraic expression which can be one of the following:

Do you remember what a monomial is? (a) Number: 3, –6 (b) Variable: x, y

Do you remember what a monomial is? (c) Product of a number and variable(s) : 3x, y, –4xy2

Good. Note that expressions like and are not monomials. 1 x 4 w 2 x 4 w (c) Product of a number and variable(s) : 3x, y, –4xy2

Do you remember what a polynomial is? A polynomial can be a monomial or the sum of two or more monomials.

Do you remember what a polynomial is? For example, 3bc, 4 + 2a , 5 + a + 2c are polynomials.

The following are some key terminologies of polynomials: Degree of the polynomial : The highest degree of all its term(s). Constant term: The term that does not contain a variable. Coefficient of a term: The numerical part of the term.

Degree of the polynomial = 3 For example, Degree of the polynomial = 3 Constant term = –7 4x2 – 6x3 – 7 Coefficient of x3 = –6 Coefficient of x2 = 4 What is the coefficient of x ?

Degree of the polynomial = 3 For example, Degree of the polynomial = 3 Constant term = –7 4x2 – 6x3 – 7 Coefficient of x3 = –6 Coefficient of x2 = 4 The coefficient of x is 0, i.e. 4x2 – 6x3 + 0x – 7.

Degree of the polynomial = 3 For example, Degree of the polynomial = 3 Constant term = –7 4x2 – 6x3 – 7 Coefficient of x3 = –6 Coefficient of x2 = 4 Coefficient of x = 0

Degree of the polynomial = For example, Degree of the polynomial = 3 Constant term = 8 5 x + 8 – 2 x 3 Coefficient of x3 = –2 Coefficient of x2 = Coefficient of x = 5

We usually arrange the terms of a polynomial in descending powers or ascending powers of a variable. e.g. For the polynomial 2x + 5x3 – 4 + 3x2 , we have: (i) Arrange in descending powers of x: 5x3 + 3x2 + 2x – 4 (ii) Arrange in ascending powers of x: –4 + 2x + 3x2 + 5x3

Grouping the like terms. Addition and Subtraction of Polynomials Addition and subtraction of polynomials can be performed by combing like terms after removing the brackets. Addition of polynomials: (4x + 3) + (x – 6) Remove the brackets, .i.e. +(x – 6) = + x – 6. = 4x + 3 + x – 6 = 4x + x + 3 – 6 Grouping the like terms. = 5x – 3

Same result! Subtraction of polynomials: (6x2 + 2x) – (4x2 + x) We can also simplify the above expression as follows: 6x2 + 2x 6x2 – 4x2 2x – x –) 4x2 + x Same result! 2x2 + x

Follow-up question (a) Simplify (x2 – 5x + 1) – (3x2 – 6x + 4). (a) Alternative Solution –) 3x2 – 6x + 4 x2 – 5x + 1 –2x2 + x – 3 Line up the like terms in the same column.

Follow-up question (cont’d) (b) Simplify (1 + x2 – 2x) + (2 – 3x2 + 4x) and arrange the terms in descending powers of x. (b) (1 + x2 – 2x) + (2 – 3x2 + 4x) = 1 + x2 – 2x + 2 – 3x2 + 4x = x2 – 3x2 – 2x + 4x + 1 + 2 = –2x2 + 2x + 3 Alternative Solution +) –3x2 + 4x + 2 x2 – 2x + 1 –2x2 + 2x + 3 Arrange the terms in descending powers of x.

Multiplication of Polynomials We can perform multiplication by applying the distribution law of multiplication: a(x + y) = ax + ay or (x + y)a = xa + ya e.g. (m + 1)(m + 2) = (m + 1)m + (m + 1)2 = m2 + m + 2m + 2 (x + y) a = xa + ya = m2 + 3m + 2

We can also do the expansion by using the method of long multiplication. Line up the like terms in the same column. m + 1 ×) m + 2 m2 + m +) 2m + 2 m2 + 3m + 2

Follow-up question Expand (2 – a2 + 3a)(–2a + 4), and arrange the terms in descending powers of a. (2 – a2 + 3a)(–2a + 4) =(2 – a2 + 3a)(–2a) + (2 – a2 + 3a)(4) = –4a + 2a3 – 6a2 + 8 – 4a2 + 12a –a2 + 3a + 2 ×) –2a + 4 2a3 – 10a2 + 8a + 8 – 4a 2a3 Alternative Solution – 6a2 + 8 +) – 4a2 + 12a = 2a3 – 6a2 – 4a2 – 4a + 12a + 8 = 2a3 – 10a2 + 8a + 8