What this chapter is about
This chapter introduces you to polynomials, which are algebraic expressions built from variables and constants using addition, subtraction and multiplication. You will learn to identify different types of polynomials based on the number of terms they have and the highest power of the variable they contain. The focus is on linear polynomials, where the highest power of the variable is one.
Understanding polynomials is essential because they appear throughout higher mathematics. They help us model real-world situations, from calculating costs to describing motion. Before studying quadratic and higher-degree polynomials, you must be comfortable with the simplest kind: linear polynomials.
After studying this chapter, you should be able to recognise polynomials, classify them by degree and number of terms, find the value of a polynomial at a given point, and determine the zero of a linear polynomial.
Key ideas
- A polynomial in one variable x is an expression of the form a₀ + a₁x + a₂x² + ... + aₙxⁿ, where a₀, a₁, a₂, ... are constants (called coefficients) and n is a whole number.
- The degree of a polynomial is the highest power of the variable with a non-zero coefficient. For example, 5x³ + 2x − 7 has degree 3.
- A linear polynomial has degree 1 and takes the general form ax + b, where a ≠ 0. Examples: 3x + 4, −2x + 1, x − 5.
- Polynomials are named by the number of terms: a monomial has one term, a binomial has two terms, and a trinomial has three terms.
- The value of a polynomial p(x) at x = k is found by substituting k for x. This value is written as p(k).
- A zero of a polynomial p(x) is a number k such that p(k) = 0. For a linear polynomial ax + b, the zero is x = −b/a.
- Expressions like 1/x, √x, or x^(−2) are not polynomials because they involve negative or fractional powers of the variable.
Formulas and facts to remember
- Formula or fact: General form of linear polynomial: ax + b, where a ≠ 0 · Meaning: A polynomial with degree exactly 1
- Formula or fact: Zero of ax + b is x = −b/a · Meaning: The value of x that makes the polynomial equal to zero
- Formula or fact: Degree of a constant polynomial (non-zero): 0 · Meaning: A number like 7 or −3 is a polynomial of degree zero
- Formula or fact: Degree of zero polynomial: not defined · Meaning: The polynomial 0 has no defined degree
- Formula or fact: p(k) means substitute x = k in p(x) · Meaning: How to evaluate a polynomial at a specific point
Worked examples
Example 1: Classify a polynomial and find its degree
Consider the expression 4x² − 3x + 6.
Step 1: Check if it is a polynomial. All powers of x are whole numbers (2, 1, 0), so yes, it is a polynomial.
Step 2: Find the degree. The highest power of x is 2, so the degree is 2.
Step 3: Count the terms. There are three terms: 4x², −3x, and 6. So it is a trinomial.
Answer: It is a polynomial of degree 2 (a quadratic trinomial).
---
Example 2: Find the value of a polynomial at a given point
Let p(x) = 2x + 5. Find p(3).
Step 1: Substitute x = 3 into the polynomial. p(3) = 2 × 3 + 5
Step 2: Calculate. p(3) = 6 + 5 = 11
Answer: p(3) = 11.
---
Example 3: Find the zero of a linear polynomial
Find the zero of the polynomial p(x) = 3x − 9.
Step 1: Set p(x) = 0. 3x − 9 = 0
Step 2: Solve for x. 3x = 9 x = 9/3 x = 3
Step 3: Verify by substituting x = 3. p(3) = 3 × 3 − 9 = 9 − 9 = 0 ✓
Answer: The zero of 3x − 9 is x = 3.
Common mistakes
- Thinking x⁻¹ or 1/x is a polynomial → it is not, because the power of x must be a non-negative whole number.
- Confusing degree with number of terms → degree is about the highest power, not how many terms exist.
- Writing the zero of ax + b as b/a instead of −b/a → always take the negative of b divided by a.
- Believing every algebraic expression is a polynomial → expressions with √x or x in the denominator are not polynomials.
- Forgetting that a non-zero constant like 7 is also a polynomial (of degree 0) → constants are polynomials with no variable term.
Quick revision
- A polynomial has only whole-number powers of the variable.
- Degree = highest power of x with a non-zero coefficient.
- Linear polynomial: ax + b (a ≠ 0), degree 1.
- Zero of ax + b is found by solving ax + b = 0, giving x = −b/a.
- To find p(k), substitute k wherever x appears and simplify.