SS2 First Term Further Mathematics Lesson Note – Polynomials I

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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:

a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀

where the highest power of the variable n is the degree of the polynomial; the numerical constants a_n, a_{n-1, …} a_2, 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) = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀ and

Q(x) = b_nxⁿ + b_n-1x^n-1 + … + b₂x² + b₁x + b₀

are said to be equal if:

a_n = b_n

a_n-1 = b_n-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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