Generic Rectangles
Generic rectangles (similar to the area model) provide another strategy for multiplying polynomials.
Consider \((13)(426)\), we can think of this as \((10+3)(400+20+6)\)
| \(10\) | \(3\) | |
|---|---|---|
| \(400\) | \(4000\) | \(1200\) |
| \(20\) | \(200\) | \(60\) |
| \(6\) | \(60\) | \(36\) |
Combining the parts: 6556
Generic rectangles can be used to multiply a monomial by a binomial:
Ex. \(3x(2x+7) = 6x^2+21x\)
| \(2x\) | 7 | |
|---|---|---|
| \(3x\) | \(6x^2\) | \(21x\) |
They can also be used to multiply any polynomial by any polynomial such as the trinomial by binomial below:
\((x^2+3x+2)(x-1)\)
| \(x\) | \(-1\) | |
|---|---|---|
| \(x^2\) | \(x^3\) | \(-x^2\) |
| \(3x\) | \(3x^2\) | \(-3x\) |
| \(2\) | \(2x\) | \(-2\) |
\(x^3+3x^2+2x-x^2-3x-2 = x^3+2x^2-x-2\). Notice that we need to add the red together as well as the purple to combine like terms. Compare this again to the distributive property. This method is a means of organizing your multiplication.
