Polynomials

1Summary 重點摘要

A polynomial is an expression with x combined by +−×÷. Like terms can be combined, multiplication distributes term by term.

2Vocabulary cards Vocabulary

3Step by step Steps

1. What a polynomial is, and its degreeA polynomial is an algebraic expression built from numbers and letters joined by addition and subtraction; each part separated by a plus or minus sign is called a term. For example, 3x² + 2x − 5 has three terms: 3x² is the quadratic term, 2x the linear term, and −5 the constant term. The degree of a polynomial is decided by the highest power, so this one is a quadratic polynomial. Identifying terms and degrees is the starting point of every polynomial operation, since combining, multiplying and factorising all begin by reading each term's structure.
2. Combining like termsLike terms are terms whose letter parts are exactly the same — same variable raised to the same power. To combine like terms, add or subtract only the coefficients and keep the letter part unchanged. In 3x² + 2x + 5x² − x, the terms 3x² and 5x² combine to 8x², and 2x and −x combine to x, so the expression simplifies to 8x² + x. A common mistake is treating 3x² and 2x as like terms and writing 5x³; terms with different powers can never merge, just as 2 oranges and 3 apples never become 5 oranges.
3. Adding and subtracting polynomialsTo add or subtract polynomials, remove the brackets first, then combine like terms. For (3x² + 2x) + (5x² − x + 1), removing brackets gives 3x² + 2x + 5x² − x + 1; combining x² terms gives 8x², the x terms give x, and the constant 1 stays, so the result is 8x² + x + 1. If a minus sign stands before a bracket, as in (3x² + 2x) − (5x² − x + 1), every term inside must change sign: 3x² + 2x − 5x² + x − 1 = −2x² + 3x − 1. The commonest error is flipping only the first term's sign and forgetting the rest.
4. Multiplying a monomial by a polynomialWhen multiplying a monomial by a polynomial, use the distributive law: multiply the monomial by every term inside the bracket, not just the first. For example, 2x(x + 3) = 2x × x + 2x × 3 = 2x² + 6x. Also, −3x(2x − 4) = −6x² + 12x, since a negative times a negative is positive. The classic mistake is multiplying only the first term and stopping at 2x × x = 2x²; count the terms inside the bracket first, and the expansion must contain just as many (except for terms that merge).
5. Two binomials: the FOIL methodTo multiply two binomials, use the FOIL method: First, Outer, Inner, Last, then combine like terms among the four products. For (x + 2)(x + 3): First x × x = x²; Outer x × 3 = 3x; Inner 2 × x = 2x; Last 2 × 3 = 6; combining gives x² + 5x + 6. Common errors are dropping the two middle terms and writing only x² + 6, or forgetting to add 3x and 2x; remember that before combining, the expansion must show four terms.
6. The general method for larger productsWhen degrees are higher or there are more terms, FOIL is not enough; multiply every term of the first polynomial by every term of the second, then combine like terms. For (x + 1)(x² + 2x + 3): x times the three terms gives x³ + 2x² + 3x, then 1 times them gives x² + 2x + 3; combining yields x³ + 3x² + 5x + 3. Check the count: 2 terms times 3 terms must produce 6 before combining; a different number means a term was lost. This term-by-term discipline is the basic skill for cubic and higher products.
7. How to check your answerAfter expanding or simplifying, the most reliable check is substitution. Choose a simple number such as x = 1, evaluate both the original expression and the result, and the two values must agree. To verify (x + 2)(x + 3) = x² + 5x + 6, substitute x = 1: the left side gives (1 + 2)(1 + 3) = 12 and the right side 1 + 5 + 6 = 12, confirming the expansion. If the values differ, re-check step by step; substituting 0 or 1 is quickest, though a single value is only a spot check — a formal proof still compares term by term.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Only like terms can combine
• Distributive law: multiply every term
• FOIL: First, Outer, Inner, Last
• Negative × negative = positive
• Substitute a number to check