Special Products, Binomial Formulas and Pascal’s Triangle
Special products are patterns for multiplying polynomials quickly, such as the binomial formulas (a + b)2 = a2 + 2ab + b2 and (a + b)(a − b) = a2 − b2. In brief: (x + 5)2 = x2 + 10x + 25, (x − 4)2 = x2 − 8x + 16 and (x + 3)(x − 3) = x2 − 9. For higher powers, the coefficients come from Pascal’s triangle.
Knowing these patterns saves time in expansion and is essential for factoring, since each formula can be read backwards. They also help with mental arithmetic: 472 = (50 − 3)2 = 2500 − 300 + 9 = 2209. This guide lists the formulas, shows why they work, explains the cubic formulas and uses Pascal’s triangle for the expansion of (a + b)n.
What are the binomial formulas for squares?
Squaring a binomial gives a trinomial with a middle term of twice the product:
- (a + b)2 = a2 + 2ab + b2
- (a − b)2 = a2 − 2ab + b2
Why: (a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2. Geometrically, a square with side a + b is divided into a square a2, a square b2 and two rectangles ab.
Example 1
(3x + 4)2 = (3x)2 + 2 · 3x · 4 + 42 = 9x2 + 24x + 16.
Example 2
(2x − 5y)2 = 4x2 − 20xy + 25y2.
Example 3: arithmetic
1032 = (100 + 3)2 = 10,000 + 600 + 9 = 10,609. And 982 = (100 − 2)2 = 10,000 − 400 + 4 = 9,604.
What is the difference of squares formula?
(a + b)(a − b) = a2 − b2. The middle terms cancel: −ab + ab = 0. It is useful for expansion, for factoring a difference of squares and for rationalizing denominators.
Example 4
(2x + 7)(2x − 7) = 4x2 − 49. And 52 · 48 = (50 + 2)(50 − 2) = 2500 − 4 = 2496.
Example 5: factoring
x4 − 81 = (x2)2 − 92 = (x2 + 9)(x2 − 9) = (x2 + 9)(x + 3)(x − 3). Note that a sum of squares x2 + 9 does not factor over the real numbers.
What are the formulas for cubes?
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a − b)3 = a3 − 3a2b + 3ab2 − b3
- a3 + b3 = (a + b)(a2 − ab + b2)
- a3 − b3 = (a − b)(a2 + ab + b2)
Example 6
(x + 2)3 = x3 + 6x2 + 12x + 8. And x3 − 27 = (x − 3)(x2 + 3x + 9). Check with x = 4: 64 − 27 = 37 and (1)(16 + 12 + 9) = 37 ✓.
The quadratic factor in a sum or difference of cubes does not factor further over the real numbers.
How does Pascal’s triangle give the coefficients?
The expansion of (a + b)n has n + 1 terms. The powers of a go down from n to 0 and the powers of b go up from 0 to n. The coefficients are the numbers in row n of Pascal’s triangle. Each number in the triangle is the sum of the two numbers above it, and each row starts and ends with 1.
Row 4 is 1, 4, 6, 4, 1, so (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4. For (a − b)n the signs alternate. Example: (x − 1)5 uses row 5 (1, 5, 10, 10, 5, 1): x5 − 5x4 + 10x3 − 10x2 + 5x − 1.
The numbers in row n are the binomial coefficients C(n, k) = n!/(k!(n − k)!), which also count combinations.
Pascal’s triangle table, rows 0 to 8
| n | Coefficients | Sum of row |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1, 1 | 2 |
| 2 | 1, 2, 1 | 4 |
| 3 | 1, 3, 3, 1 | 8 |
| 4 | 1, 4, 6, 4, 1 | 16 |
| 5 | 1, 5, 10, 10, 5, 1 | 32 |
| 6 | 1, 6, 15, 20, 15, 6, 1 | 64 |
| 7 | 1, 7, 21, 35, 35, 21, 7, 1 | 128 |
| 8 | 1, 8, 28, 56, 70, 56, 28, 8, 1 | 256 |
The sum of row n is 2n, which follows from substituting a = b = 1 into the expansion.
Where do these formulas come from?
The formula for the square of a sum appears in Euclid’s Elements (Book II) in a geometric form. The triangle of binomial coefficients was known in India, Persia and China long before Blaise Pascal, who studied it in the seventeenth century and gave it its Western name; in China it is often named after Yang Hui, and in Iran after Omar Khayyam. The general binomial theorem for any exponent was developed by Isaac Newton.
Where are they used today?
Expansions help to simplify expressions in algebra and calculus, for instance when computing derivatives from the definition. In probability the binomial coefficients give the number of ways to get k successes in n trials. Mental calculation of squares near round numbers uses these formulas, and in computer science the coefficients are used in algorithms for polynomial arithmetic.
Which mistakes are common?
Forgetting the middle term
(a + b)2 is not a2 + b2. For a = 3, b = 4: 49 versus 25.
Wrong sign in the last term
(a − b)2 ends with + b2, because (−b)2 is positive.
Forgetting to square the coefficient
(3x + 1)2 starts with 9x2, not 3x2.
Mixing up the cube formulas
The factor a2 − ab + b2 goes with the sum a3 + b3, and a2 + ab + b2 goes with the difference a3 − b3.