A binomial is the sum of two terms: `(x + 3)` and `(x + 2)` are both binomials. Expressions can contain binomials that will be multiplied together: `(x + 3)(x + 2)`. Each of the terms in the first binomial must be multiplied by each of the terms in the second binomial. There are various methods for doing this (such as FOIL and FACE) but one of the simpler ways is to use a grid:
| `x` | `+3` | |
| `x` | ||
| `+2` |
Place one expression `(x + 3)` in the top row and the other `(x + 2)` as a column.
Carry out the multiplication of each term:
| `x` | `+3` | |
| `x` | `x^2` | `+3x` |
| `+2` | `+2x` | `+6` |
Add the terms together: `x^2 + 3x + 2x + 6`;
and finally collect the like terms: `x^2 + 5x + 6`.
Expanding the product of two binomials is the opposite of factorising a quadratic.
Expand and simplify: `(2x - 3)(5x + 7)`.
`10x^2 -15x + 14x -21 = 10x^2 - x + 21`
| `2x` | `-3` | |
| `5x` | `10x^2` | `-15x` |
| `+7` | `14x` | `-21` |
Answer: `10x^2 - x - 21`
Expand and simplify: `(ab + 3)(4b + 2)`.
`4ab^2 +12b + 2ab +6`; the expression cannot be simplified any further.
| `` | `ab` | `+3` |
| `4b` | `4ab^2` | `+12b` |
| `+2` | `+2ab` | `+6` |
Answer: `4ab^2 + 12b + 2ab + 6`