Multiplying Polynomials

1Summary 重點摘要

Learn multiplying polynomials, from simplest monomial times polynomial, to binomial times binomial (FOIL).

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Monomial times polynomialThe simplest multiplication uses the distributive law: multiply the outside monomial into every term of the bracket. For example, 2x(3x + 4) = 2x×3x + 2x×4 = 6x² + 8x, multiplying coefficients together and x by x to get x². After expanding, count the terms — two terms inside means two terms out; a mismatch signals a missed multiplication. This kind of product underpins all the later rules, since FOIL is really the distributive law applied twice.
2. Step 2: The FOIL methodThe most common pattern is binomial times binomial, (a+b)(c+d), expanded systematically with the FOIL mnemonic: First multiplies the leading terms, Outer the two outside terms, Inner the two inside terms, and Last the two trailing terms. None of the four products may be skipped — that is the safeguard against missing terms. FOIL is in essence the distributive law used twice; following the fixed order like a scan, and drawing arcs over the four products while learning, builds a reliable habit.
3. Step 3: A complete exampleTake (x + 2)(x + 3): First = x·x = x², Outer = x·3 = 3x, Inner = 2·x = 2x, Last = 2·3 = 6, and adding gives x² + 3x + 2x + 6, which combines to x² + 5x + 6. The worked example (2x - 1)(x + 4) follows the same four steps to get 2x² + 8x - x - 4 = 2x² + 7x - 4. Afterwards test with a number: substituting x = 1 into both the original product and the answer gives 5 in each case, confirming the expansion.
4. Step 4: Handle negative signs carefullyWith negative signs, apply the sign rules carefully: positive times positive is positive, positive times negative is negative, negative times negative is positive. For example, (x - 2)(x + 3) = x² + 3x - 2x - 6, where -2 times 3 gives -6 rather than +6, and combining yields x² + x - 6. The two usual errors are computing -2 × 3 as +6, and slipping when combining 3x - 2x; circling each signed product, deciding the sign first and the number second, greatly reduces such mistakes.
5. Step 5: Combine like termsAfter the four products are written, the Outer and Inner terms are usually like terms and must be combined — in the example, 3x + 2x becomes 5x — while the x² and constant terms generally have no partners and are copied across. Never present the answer uncombined; this is a common way to lose marks. Finally arrange in descending powers and count the terms: a binomial times a binomial gives four terms before combining and usually three after.
6. Step 6: Advice for examinationsIn examinations, use FOIL horizontally for simple products; when expressions grow longer, the vertical method is safer — align the two expressions as in integer multiplication, multiply term by term, then add, which avoids omissions and is easy to check. Whichever method is chosen, substitute a simple value such as x = 1 or x = 2 afterwards to confirm the original and the answer agree. With time to spare, re-check the signs once more, especially the negative terms; these habits protect most of the marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Monomial times polynomial use distributive law
• Binomial times binomial use FOIL
• First, Outer, Inner, Last
• Be careful with negative signs
• Finally combine like terms