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WEEK ONE
TOPIC: POLYNOMIALS 1
SUB-TOPICS:
(a) Definition of polynomials.
(b) Basic operations on polynomials.
(c) Remainder and factor theorem.
(d) Zeros of polynomials.
SUB-TOPIC 1
Definition of Polynomials
A polynomial is a mathematical expression which is a sum of terms, each term consisting a variable or variables raised to a power and multiplied by a coefficient. A polynomial of one variable x (univariate) has the following as its general form:
anxn + an-1xn-1 + … + a2x2 + a1x + a0
where the highest power of the variable n is the degree of the polynomial; the numerical constants an, an-1, … a2, a1 are called the coefficients of the polynomial, while a0 is called the constant term.
Examples of polynomials include:
- 3x2 – 2x + 4
- 2x3 + 3x2 + 5x + 3
A function whose values are given by a polynomial is called a polynomial function. Eg: f(x) = 2x3 + 3x2 + 5x + 3
An equation that is obtained when we set a polynomial equal to zero is called a polynomial equation. E.g.: 2x3 + 3x2 + 5x + 3 = 0
Equality of polynomials
Two polynomials,
P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 and
Q(x) = bnxn + bn-1xn-1 + … + b2x2 + b1x + b0
are said to be equal if:
an = bn
an-1 = bn-1
a2 = b2
a1 = b1
a0 = b0
The value that is obtained by substituting a for x in a polynomial P(x) is denoted by P(a).
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