## Multiplication of Polynomials

• The general rule is that each term in the first factor has to multiply each term in the other factor
• The number of products you get has to be the number of terms in the first factor times the number of terms in the second factor. For example, a binomial times a binomial gives four products, while a binomial times a trinomial gives six products.
• Be very careful and methodical to avoid missing any terms
• After the multiplication is complete you can try to collect like terms to simplify the result

### Example: Product of a binomial and a trinomial

(x + 2)(x2 2x + 3)

There are six possible products. We can start with the x and multiply it by all three terms in the other factor, and then do the same with the 2. It would look like this:

(x+ 2)(x2 2x + 3)
=(x)x2(x)2x + (x)3 + (2)x2(2)2x + (2)3
= x3 2x2 + 3x + 2x2 4x + 6
= x3 x + 6

This method can get hard to keep track of when there are many terms. There is, however, a more systematic method based on the stacked method of multiplying numbers:

 Stack the factors, keeping like degree terms lined up vertically: Multiply the 2 and the 3: Multiply the 2 and the –2x: Multiply the 2 and the x2: Now multiply the x by each term above it, and write the results down underneath, keeping like degree terms lined up vertically:   Then you just add up the like terms that are conveniently stacked above one another: This stacked method is much safer, because you are far less likely to accidentally overlook one of the products, but it does take up more space on the paper.

### Product of a monomial and a binomial: Distributive Law

Example: ab(2a + 1) = ab(2a) + ab(1) = 2a2b + ab

### Product of two binomials: FOIL (First-Outer-Inner-Last)

Because the situation of a binomial times a binomial is so common, it helps to use a quick mnemonic device to help remember all the products. This is called the FOIL method.

Example: 1.     The F stands for first, which means the x in the first factor times the x in the second factor

2.     The O stands for outer, which means the x in the first factor times the 3 in the second factor

3.     The I stands for inner, which means the 2 in the first factor times the x in the second factor

4.     The L stands for last, which means the 2 in the first factor times the 3 in the second factor

·       Of course you would then combine the 3x + 2x into a 5x, because they are like terms, so the final result is