Multiplying polynomials is not a difficult thing to understand, but it can be difficult to execute correctly. Careful attention to detail, and neatly organized work, are both crucial to consistently correct work. Seriously! This isn’t your English teacher yelling at you about penmanship. This is saying that the physical structure and spatial arrangement, on paper, will be a key factor in the success, or failure, you experience with this important topic.
There are a few ways to approach these issues that will be shared in the notes section below. Read through each tab, take notes, try the examples on your own. Then, watch the videos and try the practice problems.
In the previous section we learned how to set up and distribute the multiplication of two binomials. When you see something like (x + 3)(x + 4), what is happening is that every term in the first group, (x + 3), must be distributed to the second binomial (x + 4). We set that up as x(x + 4) + 3(x + 4). We distribute each, and then, if possible, combine like terms.
The only thing that will be different about what you’ll learn here is the number of terms being multiplied, and some tricks you can use to help make that more manageable. Let’s see an example.
This is example 1. We will look at another way of doing this same problem later in this section.
Just like before, we have to multiply x and -3 with the trinomial (which is quadratic because the degree is two), x2 + 4x – 1. We can set that up systematically as follows.
Now, distribute each to get the following. Be careful with the signs, that’s the most common mistake!
We have this jumbled up, messy polynomial expression. Some terms are out of order (descending order), and we have like terms. There is a + 4x2 and a – 3x2, as well as a –x and a – 12x. Let’s put these together and write our final product in descending order.
Not too bad, if you’re careful and patient. Let’s look at one more example, here with two trinomials. The process and meaning are the same, regardless of how large, or small, the polynomials are. We distribute and combine like terms, writing our final product in descending order.
Our set up and process will not change just because there are more terms. That’s why we spent the time learning to do this long-hand. It always works. The short-cuts only work in specific situations. Those specific situations are often worth exploring deeply, and the short-cuts a good use of working memory. However, short-cuts are forgotten and confused, or misapplied. The fundamentals are always important. Let’s do this example now.
Without doubt you can probably imagine how just being careful with the signs and coefficients and exponent values is a bit of a nightmare. No single part is difficult, but your patience and care, or lack of, will come into play here.
There’s another method you can use, if you like. It is similar to column multiplication, like this below.
Here, you’d multiply the four times the 1, then the 0, then the three, writing each value in the column below the upper number. You’d get this.
Then, you’d take a place a zero under the four, before multiplying by 1 (because it’s a 10, not a 1).
Then, you’re multiply the 201 by 1, writing each product in the column below.
Then, you’d add the columns.
Let’s set up our first example in the same fashion and see how this works. This is a little tricky to set up the first time, but once you see how it works it offers a few advantages. It is faster, takes less writing and your like terms are aligned!
Multiply the negative 3 with each term, be careful of your signs and write neatly, leaving the columns aligned vertically as you write. Try it as you read through this example.
Just like before, use your place marker.
Now multiply the x with the trinomial. Once again, be careful with signs and vertical alignment.
Now just add the columns.
Let’s set up an example where people can get confused. In the following example, which was our second of the day, do you see how the second trinomial has a 0x term? Of course you don’t literally see it, it is not there because we don’t write + 0, there’s no need. However, this is similar to multiplying by 201, which is really 2 × 100 + 0 × 10 + 1 × 1. We would not write 0 × 10, because it is just zero. But, we use the zero as a space holder when we write the entire number compressed together. The same holds true here.
We would prefer to have the longer polynomial on the top row, as it makes the set up and process easier. So, we’d write it like this.
You try the process, checking the answer with the previously worked out answer earlier in this section.
Summary: To multiply polynomials we just need to distribute each term of the first polynomial to each term of the other polynomial.