Introduction to Polynomials

Polynomials are algebraic expressions made using numbers and variables, combined through addition, subtraction, and multiplication, where the variables have only non-negative whole number powers.

The above image shows the parts of a polynomial (3x² + 5), including the coefficient, variable, exponent (power), constant, and terms.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in it. For polynomials with more than one variable, the degree is the highest sum of powers in any term.

Example: In 3x⁴ + 7, the highest power of x is 4, so the degree is 4.

Types

Polynomials can be categorised into several types based on the number of terms they contain or their degree.

Based on the Number of Terms:

• Monomial: A polynomial with just one term. For example, 7x³ or −4.
• Binomial: A polynomial with two terms. For example, x² − 5x.
• Trinomial: A polynomial with three terms. For example, 2x² − 3x + 4.
• Multinomial: A polynomial with more than three terms. For example, x⁴ − 2x³ + 3x² − x + 1.

Based on Degree:

• Constant Polynomial: A polynomial of degree 0. It has no variable terms, just a constant. For example, 5 or −3.
• Linear Polynomial: A polynomial of degree 1. It forms a straight line when graphed. For example, 3x + 2.
• Quadratic Polynomial: A polynomial of degree 2. It forms a parabolic shape when graphed. For example, x² − 4x + 4.
• Cubic Polynomial: A polynomial of degree 3. It can have points of inflexion and typically has the shape of an S-curve when graphed. For example, x³ − 6x² + 11x − 6.

Properties and Theorems

Polynomials follow some important rules and theorems that make them easier to understand and solve:

• Degree Rule: The degree of the sum or difference of two polynomials is at most the highest degree among them, while the degree of their product is the sum of their degrees.

• deg(A ± B) ≤ max(deg A, deg B)
• deg(A × B) = deg A + deg B

• Division Algorithm: For any polynomials A and B (where B ≠ 0), we can write:

A = BQ + R, where the degree of R is less than the degree of B.

• Factor Theorem: A polynomial P(x) is divisible by (x − a) if and only if P(a) = 0.
• Remainder Theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a).
• Number of Zeros: A polynomial of degree n can have at most n zeros.
• Zeros Property: If a polynomial P is divisible by another polynomial Q, then every zero of Q is also a zero of P.

Polynomial Function

A polynomial function is a function expressed as a sum of terms in which each variable has a non-negative integer exponent, and each term has a constant coefficient.

The general form of a polynomial function is:

f(x) = a​_n x ⁿ + a_{n - 1} ​x _{n -1} + ⋯ + a₁ x + a₀​

where:

• a₀ , a₁ ​,…, a_n​ are constant coefficients.
• x is the variable.
• n is a non-negative integer representing the degree of the polynomial, provided a_n ≠ 0.

Operations on Polynomial Functions

Operations on polynomial functions involve adding, subtracting, multiplying, and dividing polynomial expressions to form new functions.

1. Addition of Polynomial Functions : To add two polynomial functions, combine their like terms.
2. Subtraction of Polynomial Functions: To subtract two polynomial functions, subtract the corresponding terms and change the signs of the terms inside the subtracted polynomial.
3. Multiplication of Polynomial Functions : To multiply two polynomial functions, multiply each term of one polynomial by every term of the other polynomial and combine like terms.
4. Division of Polynomial Functions: To divide two polynomial functions, divide one polynomial by the other, provided the denominator is not zero.

Practice Questions on Polynomials