1. Revision on Polynomials

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

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

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

5. Good. Note that expressions like and are not monomials. 1 x 4 w2 x 4 w
(c) Product of a number and variable(s) :
3x, y, –4xy2

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

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

8. 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.

9. 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 ?

10. 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.

11. 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

12. 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

13. 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

14. 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 x – 6
= 4x + x + 3 – 6
Grouping the like terms.
= 5x – 3

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

16. 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.

17. 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
= –2x2 + 2x + 3
Alternative Solution
+) –3x2 + 4x + 2
x2 – 2x + 1
–2x2 + 2x + 3
Arrange the terms in descending powers of x.

18. 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

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

20. 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