Introduction to Polynomials

1Summary 重點摘要

Learn what polynomials are, and basic concepts like term, coefficient, degree.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What a polynomial isA polynomial is an algebraic expression made of variables (such as x) and constants joined by addition and subtraction, for example 3x² + 2x - 5. Two rules define it: the variable never appears in a denominator and never carries a negative index, so 4/x and 2x⁻¹ are not polynomials, and neither is an expression with a fractional index such as √x. Mastering this definition lets you classify any expression at first glance, which underpins the whole topic.
2. Step 2: Understanding termsEach part of a polynomial separated by a plus or minus sign is called a term: 3x² + 2x - 5 has three terms, namely 3x², 2x and -5. The sign in front of a term belongs to the term itself and must travel with it when terms move or merge — the term is -5, never +5. A term containing only a number is a constant term, and a polynomial with just one term is a monomial. Counting terms carefully is the first step before simplifying or finding the degree.
3. Step 3: Understanding coefficientsThe numerical factor written in front of the variable of a term is its coefficient: the coefficient of 3x² is 3, and of 2x it is 2. When no number is written, the coefficient is understood to be 1, so x means 1x, while the coefficient of -x is -1 — the two cases beginners overlook most often. A constant term can be viewed as a term whose variable part is x⁰, and its number is the coefficient. Once coefficients are read correctly, combining like terms becomes simple arithmetic on them.
4. Step 4: Understanding degreeThe degree of a polynomial is the highest power of the variable among all its terms: 3x² + 2x - 5 has degree 2. A constant term has degree 0, since 5 can be written as 5x⁰. Check every term rather than reading only the first: 2x³ + x⁵ has degree 5, not 3, and degree is not the same as the number of terms. Simplify fully before naming the degree, because examinations often ask for it directly.
5. Step 5: Identifying like termsLike terms are terms whose variable parts are identical — same letter and same power — while the coefficients may differ. For example, 3x² and 5x² are like terms, as are 2x and -7x, but 3x² and 2x are not, because the powers differ. Only the variable part matters; signs and positions are irrelevant. All constant terms are like terms of one another, such as 6 and -5. Underlining or circling like terms first is a habit that prevents omissions and mismatches when simplifying.
6. Step 6: Combining like termsThe most basic simplification of a polynomial is combining like terms: add or subtract the coefficients and copy the variable part. For example, 3x² + 2x + 5x² - 3x = (3+5)x² + (2-3)x = 8x² - x. Unlike terms can never be merged, so 3x² + 2x is already in simplest form. Arrange the result in descending order of powers for clear presentation, and verify by substituting a value such as x = 1 into both the original expression and the answer, exactly as the worked example checks 3x² + 2x + 5.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Polynomial = variables and constants added/subtracted
• Term = each part, with sign
• Coefficient = number in front of variable
• Degree = highest power
• Only like terms can combine