Learning Objectives
- Understand polynomials in one variable and their degree
- Classify polynomials based on degree and number of terms
- Find the value and zeroes of a polynomial
- Apply the Remainder Theorem and Factor Theorem
- Factorise polynomials using algebraic identities
Key Concepts
Polynomials in One Variable
A polynomial in variable x is an expression of the form a₀ + a₁x + a₂x² + ... + aₙxⁿ, where a₀, a₁, ..., aₙ are real numbers and n is a non-negative integer.
Degree: The highest power of the variable in the polynomial.
Classification by Degree
- Linear Polynomial: Degree 1 (e.g., 2x + 3)
- Quadratic Polynomial: Degree 2 (e.g., x² + 5x + 6)
- Cubic Polynomial: Degree 3 (e.g., x³ - 2x² + x - 1)
Classification by Number of Terms
- Monomial: One term (e.g., 5x²)
- Binomial: Two terms (e.g., x + 3)
- Trinomial: Three terms (e.g., x² + 2x + 1)
Zeroes of a Polynomial
A real number k is a zero of polynomial p(x) if p(k) = 0. A linear polynomial ax + b has exactly one zero: x = -b/a. A polynomial of degree n can have at most n zeroes.
Remainder Theorem
If p(x) is a polynomial of degree ≥ 1 and a is any real number, then the remainder when p(x) is divided by (x - a) is p(a).
Factor Theorem
If p(x) is a polynomial of degree ≥ 1 and p(a) = 0, then (x - a) is a factor of p(x). Conversely, if (x - a) is a factor of p(x), then p(a) = 0.
Important Algebraic Identities
- (x + y)² = x² + 2xy + y²
- (x - y)² = x² - 2xy + y²
- x² - y² = (x + y)(x - y)
- (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
- (x + y)³ = x³ + y³ + 3xy(x + y)
- (x - y)³ = x³ - y³ - 3xy(x - y)
- x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
Summary
Polynomials are algebraic expressions with non-negative integer exponents. They are classified by degree and number of terms. The Remainder Theorem connects polynomial division with evaluation, while the Factor Theorem provides a method for factorisation. Standard algebraic identities are powerful tools for expanding and factorising expressions.
Important Terms
- Polynomial: An algebraic expression with non-negative integer powers of a variable
- Degree: The highest exponent of the variable in the polynomial
- Zero of a Polynomial: A value of the variable for which the polynomial equals zero
- Remainder Theorem: The remainder on dividing p(x) by (x - a) equals p(a)
- Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0
Quick Revision
- Degree 1 = linear, Degree 2 = quadratic, Degree 3 = cubic
- A polynomial of degree n has at most n zeroes
- Remainder when p(x) is divided by (x - a) is p(a)
- If p(a) = 0 then (x - a) is a factor of p(x)
- Use algebraic identities for quick factorisation