Multiplying polynomials

Multiplying polynomials is easy enough, but it can get a bit messy. Especially when dealing with a few variables. But if you know what you are doing, you will manage quite nicely. So let us get down to business.

There is a simple logic behind multiplying polynomials – just multiply every term in the first polynomial with every term of the second polynomial. After that, just tidy up the remaining terms by performing the necessary mathematical operations, such as addition and subtraction. Apart from that, all other rules for multiplication and the order of operations still apply and they should be observed.

Example 1.

So, if you have a polynomial like this:

(4m + 3) * (3m – 2m)

…the first thing you should do is to multiply the terms from the first polynomial with each term in the second one. The process looks like this:

4m*3m + 4m *(-2m) + 3*3m + 3*(-2m)

12m2 – 8m2 + 9m – 6m

After a bit of tidying up, it should look like this:

4m2 + 3m

And that is it. That is the process of multiplying polynomials. It is easy, right? Now we are going to solve a bit more complicated example to show you how to deal with the clutter that appears in these cases.

Example 2.

Let us assume that we have to simplify the product of these polynomials.

(-a + 3b) * (-a2 + ab + 3b2)

Again, we have to start by multiplying each term from the first polynomial with the terms in the second one.

(-a)*(-a2) + (-a)*ab + (-a)*3b2 + 3b2*(-a2) + 3b*ab + 3b*3b2

a3 - a2b – 3ab2 – 3a2b + 3ab2 + 9b3

Now it is time to perform the addition and subtraction to get this mathematical expression in order. Keep in mind that these operations can only be performed with terms whose variables are exactly the same. We will rewrite this expression in a way that these variables are next to each other. So, we get something like this:

a3 – a2b – 3a2b – 3ab2 + 3ab2 + 9b3

We can leave out the two terms that have the same value, but opposite signs since their sum is 0. That means the result of our simplification is:

a3 – 4a2b + 9b3

As you can see, even the most complicated examples are not that difficult to solve. However, a considerable amount of concentration is required because mistakes can happen pretty easily. When you deal with multiplying polynomials, be sure to check your calculations before going further with an assignment. It is worth the extra effort.

So, this is all there is to multiplying polynomials. They can get more complicated by adding more variables or extra polynomials, but if you follow these basic rules and focus on your calculations, you can solve them all. If you wish to practice multiplying polynomials, feel free to use the worksheets below.

Multiplying polynomials exams for teachers

Exam Name File Size Downloads Upload date
Single variable – Integers
Multiplication of polynomials – Integers with a single variable – very easy 557.8 kB 1434 October 13, 2012
Multiplication of polynomials – Integers with a single variable – easy 556.4 kB 977 October 13, 2012
Multiplication of polynomials – Integers with a single variable – medium 580.7 kB 1509 October 13, 2012
Multiplication of polynomials – Integers with a single variable – hard 567.1 kB 1260 October 13, 2012
Multiplication of polynomials – Integers with a single variable – very hard 589.3 kB 1421 October 13, 2012
Single variable – Decimals
Multiplication of polynomials – Decimals with a single variable – very easy 570.4 kB 562 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – easy 573.5 kB 404 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – medium 610.6 kB 472 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – hard 600.3 kB 444 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – very hard 642.1 kB 476 October 13, 2012
Single variable – Fractions
Multiplication of polynomials – Fractions with a single variable – very easy 579.5 kB 524 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – easy 587.8 kB 460 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – medium 626.1 kB 481 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – hard 633.1 kB 514 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – very hard 680.5 kB 463 October 13, 2012
Two variables – Integers
Multiplication of polynomials – Integers with two variables – very easy 560 kB 860 October 13, 2012
Multiplication of polynomials – Integers with two variables – easy 559.8 kB 586 October 13, 2012
Multiplication of polynomials – Integers with two variables – medium 593.7 kB 894 October 13, 2012
Multiplication of polynomials – Integers with two variables – hard 583.5 kB 768 October 13, 2012
Multiplication of polynomials – Integers with two variables – very hard 618.7 kB 767 October 13, 2012
Two variables – Decimals
Multiplication of polynomials – Decimals with two variables – very easy 582.8 kB 385 October 13, 2012
Multiplication of polynomials – Decimals with two variables – easy 581 kB 431 October 13, 2012
Multiplication of polynomials – Decimals with two variables – medium 625.3 kB 382 October 13, 2012
Multiplication of polynomials – Decimals with two variables – hard 624.3 kB 445 October 13, 2012
Multiplication of polynomials – Decimals with two variables – very hard 670.5 kB 426 October 13, 2012
Two variables – Fractions
Multiplication of polynomials – Fractions with two variables – very easy 591 kB 456 October 13, 2012
Multiplication of polynomials – Fractions with two variables – easy 599.3 kB 400 October 13, 2012
Multiplication of polynomials – Fractions with two variables – medium 643.3 kB 441 October 13, 2012
Multiplication of polynomials – Fractions with two variables – hard 640.9 kB 494 October 13, 2012
Multiplication of polynomials – Fractions with two variables – very hard 689.5 kB 506 October 13, 2012


Multiplying polynomials worksheets for students

Worksheet Name File Size Downloads Upload date
Single variable – Integers
Integers – Simplify product of monomials and binomials 140.3 kB 1143 October 14, 2012
Integers – Simplify product of monomials and trinomials 199.7 kB 716 October 14, 2012
Integers – Simplify product of binomials 176.1 kB 1291 October 14, 2012
Integers – Simplify product of binomials and trinomials 260.4 kB 1411 October 14, 2012
Single variable – Decimals
Decimals – Simplify product of monomials and binomials 148.8 kB 448 October 14, 2012
Decimals – Simplify product of monomials and trinomials 213.1 kB 465 October 14, 2012
Decimals – Simplify product of binomials 188.2 kB 499 October 14, 2012
Decimals – Simplify product of binomials and trinomials 275.5 kB 443 October 14, 2012
Single variable – Fractions
Fractions – Simplify product of monomials and binomials 274.9 kB 475 October 14, 2012
Fractions – Simplify product of monomials and trinomials 397.9 kB 394 October 14, 2012
Fractions – Simplify product of binomials 372.3 kB 478 October 14, 2012
Fractions – Simplify product of binomials and trinomials 2.2 MB 462 October 14, 2012
Two variables – Integers
Integers – Simplify product of monomials and binomials 167.8 kB 1342 October 14, 2012
Integers – Simplify product of monomials and trinomials 257.7 kB 1120 October 14, 2012
Integers – Simplify product of binomials 241 kB 2331 October 14, 2012
Integers – Simplify product of binomials and trinomials 327.8 kB 2068 October 14, 2012
Single variable – Decimals
Decimals – Simplify product of monomials and binomials 179.2 kB 481 October 14, 2012
Decimals – Simplify product of monomials and trinomials 269.3 kB 486 October 14, 2012
Decimals – Simplify product of binomials 252.8 kB 407 October 14, 2012
Decimals – Simplify product of binomials and trinomials 346.3 kB 446 October 14, 2012
Single variable – Fractions
Fractions – Simplify product of monomials and binomials 337.2 kB 512 October 14, 2012
Fractions – Simplify product of monomials and trinomials 483.9 kB 485 October 14, 2012
Fractions – Simplify product of binomials 446 kB 409 October 14, 2012
Fractions – Simplify product of binomials and trinomials 817.7 kB 488 October 14, 2012