Words you need
A monomial is one term, like 5, 3x or −2x²y. A polynomial is a sum of terms, like x² + 5x + 6. A binomial has two terms (x + 2). The number in front is the coefficient. Like terms have exactly the same letters with the same powers: 3x and 2x are like; 3x and 3x² are not.
Before and after multiplying, write terms in order of falling power (x², then x, then the number) and add like terms.
Monomial × monomial
Multiply the numbers, then multiply the letters. For the same letter, add the powers: x^a · x^b = x^(a+b).
- 2x · 3x = 6x²
- (−4a²b)(3ab³) = −12a³b⁴
- 5y · (−2y³) = −10y⁴
Sign rule: (+)(+) = +, (−)(−) = +, (+)(−) = −.
Monomial × polynomial
Use the distributive law: a(b + c) = ab + ac. The monomial multiplies each term inside the bracket.
2x(x + 3) = 2x² + 6x. −3a(a² − 2a + 5) = −3a³ + 6a² − 15a.
In the 3D you see it as a rectangle 2x tall: one blue part (2x²) and one green part (6x).
Polynomial × polynomial
Multiply every term of the first bracket by every term of the second, then collect like terms.
(x + 2)(x + 3) = x·x + x·3 + 2·x + 2·3 = x² + 3x + 2x + 6 = x² + 5x + 6.
For two binomials, FOIL helps you not miss any product: First, Outer, Inner, Last. For longer polynomials, count: 2 terms × 3 terms = 6 products before collecting.
(2x − 1)(x² + 3x − 4) = 2x³ + 6x² − 8x − x² − 3x + 4 = 2x³ + 5x² − 11x + 4.
Dividing a polynomial by a monomial
Division undoes multiplication. Divide each term by the monomial and subtract powers: x^a ÷ x^b = x^(a−b).
(6x³ + 9x²) ÷ 3x = 2x² + 3x. Check: 3x(2x² + 3x) = 6x³ + 9x².
Adding and subtracting first
Many questions mix steps: expand, then add or subtract. A minus sign in front of a bracket changes every sign inside: −(x² − 3x + 2) = −x² + 3x − 2.
(x + 1)(x + 4) − x(x + 2) = x² + 5x + 4 − x² − 2x = 3x + 4.
Try it: paper tiles
Cut 1 big square (x by x, say 6 cm), 5 strips (6 cm by 1 cm) and 6 small 1 cm squares. Build a rectangle with sides (x + 2) and (x + 3). Did you use exactly 1 big square, 5 strips and 6 small squares? That is x² + 5x + 6. Now try (x + 1)(x + 4).
Key formulas and definitions
- x^a · x^b = x^(a+b)
- a(b + c) = ab + ac
- (a + b)(c + d) = ac + ad + bc + bd
- (x + a)(x + b) = x² + (a + b)x + ab
- (ax^m + bx^n) ÷ cx^k = (a/c)x^(m−k) + (b/c)x^(n−k)
Worked examples
1. Find 4a² × (−3a⁵).
Numbers: 4 × (−3) = −12. Letters: a² · a⁵ = a⁷. Answer: −12a⁷.
2. Expand 3x(2x − 5).
3x · 2x = 6x². 3x · (−5) = −15x. Answer: 6x² − 15x.
3. Expand (x + 4)(x + 5).
F: x·x = x². O: x·5 = 5x. I: 4·x = 4x. L: 4·5 = 20. Collect: x² + 9x + 20.
4. Expand (2y − 3)(y + 4).
2y·y = 2y². 2y·4 = 8y. −3·y = −3y. −3·4 = −12. Collect: 2y² + 5y − 12.
5. Expand (x + 2)(x² − x + 3).
x·x² = x³, x·(−x) = −x², x·3 = 3x, 2·x² = 2x², 2·(−x) = −2x, 2·3 = 6. Collect: x³ + x² + x + 6.
6. Divide (12m⁴ − 8m³ + 4m²) by 4m².
12m⁴ ÷ 4m² = 3m². −8m³ ÷ 4m² = −2m. 4m² ÷ 4m² = 1. Answer: 3m² − 2m + 1.
7. Simplify (a + 3)(a − 2) − (a − 1)².
(a + 3)(a − 2) = a² + a − 6. (a − 1)² = a² − 2a + 1. Subtract: a² + a − 6 − a² + 2a − 1 = 3a − 7.
Common mistakes
- Multiplying only the first term: 2(x + 3) is 2x + 6, not 2x + 3.
- Multiplying powers instead of adding them: x² · x³ = x⁵, not x⁶.
- Forgetting the middle term: (x + 3)² = x² + 6x + 9, not x² + 9.
- Losing a minus sign: (x − 2)(x + 5) has −2 · 5 = −10, so the last term is −10.