Polynomials

1Summary 重點摘要

Understand polynomials, learn addition, subtraction and multiplication.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The structure of a polynomialA polynomial is an algebraic expression built from terms joined by plus and minus signs. Each term has three features: a coefficient, a variable part and a degree. In 3x² + 2x - 5, the term 3x² is quadratic with coefficient 3, 2x is linear, and -5 is the constant term. The degree of the whole polynomial is the highest degree among its terms, so this one is quadratic. Students often forget the constant term or assume x² has no coefficient; in fact its coefficient is 1, and such details matter when finding degrees and simplifying.
2. Step 2: Like terms and how to combine themLike terms are terms whose variable parts are identical: same letter and same power. Only like terms can be added or subtracted. For example 3x² + 2x² = 5x² because both are quadratic terms and the coefficients simply add, while 3x² and 2x cannot be combined because their powers differ. When combining, add or subtract the coefficients and keep the variable part unchanged. A common exam error is treating x² and x as like terms; circle the variable part of every term and check them one by one before combining.
3. Step 3: Adding polynomialsTo add polynomials, remove the brackets and then collect like terms. For (3x² + 2x - 1) + (x² - 3x + 2), removing brackets gives 3x² + 2x - 1 + x² - 3x + 2. Combine 3x² with x² to get 4x², combine 2x with -3x to get -x, and combine -1 with 2 to get 1, so the sum is 4x² - x + 1. A plus sign in front of a bracket changes no signs inside, so most errors come from missing terms; check every term from the highest power downwards before writing the answer.
4. Step 4: Subtracting polynomials: flip every signWhen subtracting a polynomial, change the sign of every term inside the bracket after the minus sign before collecting like terms. For example (3x² + 2x - 1) - (x² - 3x + 2) = 3x² + 2x - 1 - x² + 3x - 2 = 2x² + 5x - 3. The classic mistake is flipping only the first term and forgetting that -3x becomes +3x and +2 becomes -2. Tick off each term as you flip it so that every term has been handled before you start combining.
5. Step 5: Multiplying polynomials: the distributive lawTo multiply two polynomials, use the distributive law: every term of the first expression multiplies every term of the second, then collect like terms. For (x + 2)(x + 3): x times x gives x², x times 3 gives 3x, 2 times x gives 2x and 2 times 3 gives 6, giving x² + 5x + 6 after collecting. Remember that coefficients multiply while powers add, so 2x × 3x = 6x². Beginners often miss one product; drawing a grid or ticking off each product confirms that all four appear.
6. Step 6: Simplify and checkAfter expanding, combine like terms and write the answer in descending order of degree. For instance 3x² + 2x + 5x + 6 should be simplified to 3x² + 7x + 6. Two checks are useful: confirm that no product was missed, and substitute a simple value such as x = 1 into both the original expression and the answer to see whether the numbers agree. This habit of verification removes many avoidable losses of marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Polynomial has several terms, each with coefficient and degree
• Only like terms can combine
• Subtraction must change all signs inside second bracket
• Multiplication uses distributive law, every term must multiply
• Remember combine like terms and simplify after