Page 24 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

1.2 Polynomial and Rational Functions

1
Polynomial functions
Consider the algebraic expression
3
f(x) = 3x + 4x – x + 5
2
This is an expression in terms of x, where the highest power of x is 3. An expression such as this is called a
4
3
function of x of degree 3. Similarly, g(x) = 7x – 5x + 2x + x – 3 is called a function of x of degree 4 as the
2
highest power of x is 4.
In general, a function of x in the form
n
P(x) ≡ a x + a x n – 1 + … + a x n – r + … + a x + a
0 1 r n – 1 n
+
where a  R and a ≠ 0, n  Z is called a polynomial function of degree n.
r
0
Polynomials of degrees 1, 2, 3 and 4 are also known as linear, quadratic, cubic and quartic functions respectively.
The value of a polynomial, P(x), when x = a is written as P(a).
For example,
if P(x) = 3x + 4x – 1
2
then when x = 1, P(1) = 3 + 4 – 1 = 6
and when x = 2, P(2) = 12 + 8 – 1 = 19

Algebraic operations on polynomials
Consider two linear polynomials f(x) and g(x), which are defined respectively as
f(x) = 5x + 4
and g(x) = 3x – 1

Notice that
f(x) + g(x) = (5x + 4) + (3x – 1)
= 5x + 4 + 3x – 1
= 8x + 3

f(x) – g(x) = (5x + 4) – (3x – 1)
= 2x + 5
f(x) · g(x) = (5x + 4)(3x – 1)
= 15x + 12x – 5x – 4
2
= 15x + 7x – 4
2

From the above results, we see that we can add, subtract or multiply two polynomials P(x) and Q(x) to obtain
a new polynomial.

The addition, subtraction and multiplication of two polynomials obey the commutative, associative and distributive
laws respectively.

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01a STPM Math T T1.indd 21 3/28/18 4:20 PM