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 1780 October 13, 2012
Multiplication of polynomials – Integers with a single variable – easy 556.4 kB 1260 October 13, 2012
Multiplication of polynomials – Integers with a single variable – medium 580.7 kB 1854 October 13, 2012
Multiplication of polynomials – Integers with a single variable – hard 567.1 kB 1573 October 13, 2012
Multiplication of polynomials – Integers with a single variable – very hard 589.3 kB 1776 October 13, 2012
Single variable – Decimals
Multiplication of polynomials – Decimals with a single variable – very easy 570.4 kB 758 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – easy 573.5 kB 640 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – medium 610.6 kB 697 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – hard 600.3 kB 635 October 13, 2012
Multiplication of polynomials – Decimals with a single variable – very hard 642.1 kB 666 October 13, 2012
Single variable – Fractions
Multiplication of polynomials – Fractions with a single variable – very easy 579.5 kB 726 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – easy 587.8 kB 665 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – medium 626.1 kB 680 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – hard 633.1 kB 703 October 13, 2012
Multiplication of polynomials – Fractions with a single variable – very hard 680.5 kB 666 October 13, 2012
Two variables – Integers
Multiplication of polynomials – Integers with two variables – very easy 560 kB 1098 October 13, 2012
Multiplication of polynomials – Integers with two variables – easy 559.8 kB 961 October 13, 2012
Multiplication of polynomials – Integers with two variables – medium 593.7 kB 1194 October 13, 2012
Multiplication of polynomials – Integers with two variables – hard 583.5 kB 1071 October 13, 2012
Multiplication of polynomials – Integers with two variables – very hard 618.7 kB 1097 October 13, 2012
Two variables – Decimals
Multiplication of polynomials – Decimals with two variables – very easy 582.8 kB 554 October 13, 2012
Multiplication of polynomials – Decimals with two variables – easy 581 kB 617 October 13, 2012
Multiplication of polynomials – Decimals with two variables – medium 625.3 kB 550 October 13, 2012
Multiplication of polynomials – Decimals with two variables – hard 624.3 kB 847 October 13, 2012
Multiplication of polynomials – Decimals with two variables – very hard 670.5 kB 740 October 13, 2012
Two variables – Fractions
Multiplication of polynomials – Fractions with two variables – very easy 591 kB 610 October 13, 2012
Multiplication of polynomials – Fractions with two variables – easy 599.3 kB 583 October 13, 2012
Multiplication of polynomials – Fractions with two variables – medium 643.3 kB 586 October 13, 2012
Multiplication of polynomials – Fractions with two variables – hard 640.9 kB 699 October 13, 2012
Multiplication of polynomials – Fractions with two variables – very hard 689.5 kB 672 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 1472 October 14, 2012
Integers – Simplify product of monomials and trinomials 199.7 kB 969 October 14, 2012
Integers – Simplify product of binomials 176.1 kB 1675 October 14, 2012
Integers – Simplify product of binomials and trinomials 260.4 kB 1810 October 14, 2012
Single variable – Decimals
Decimals – Simplify product of monomials and binomials 148.8 kB 686 October 14, 2012
Decimals – Simplify product of monomials and trinomials 213.1 kB 702 October 14, 2012
Decimals – Simplify product of binomials 188.2 kB 682 October 14, 2012
Decimals – Simplify product of binomials and trinomials 275.5 kB 615 October 14, 2012
Single variable – Fractions
Fractions – Simplify product of monomials and binomials 274.9 kB 668 October 14, 2012
Fractions – Simplify product of monomials and trinomials 397.9 kB 572 October 14, 2012
Fractions – Simplify product of binomials 372.3 kB 705 October 14, 2012
Fractions – Simplify product of binomials and trinomials 2.2 MB 693 October 14, 2012
Two variables – Integers
Integers – Simplify product of monomials and binomials 167.8 kB 1819 October 14, 2012
Integers – Simplify product of monomials and trinomials 257.7 kB 1756 October 14, 2012
Integers – Simplify product of binomials 241 kB 2997 October 14, 2012
Integers – Simplify product of binomials and trinomials 327.8 kB 2756 October 14, 2012
Single variable – Decimals
Decimals – Simplify product of monomials and binomials 179.2 kB 686 October 14, 2012
Decimals – Simplify product of monomials and trinomials 269.3 kB 792 October 14, 2012
Decimals – Simplify product of binomials 252.8 kB 741 October 14, 2012
Decimals – Simplify product of binomials and trinomials 346.3 kB 670 October 14, 2012
Single variable – Fractions
Fractions – Simplify product of monomials and binomials 337.2 kB 750 October 14, 2012
Fractions – Simplify product of monomials and trinomials 483.9 kB 737 October 14, 2012
Fractions – Simplify product of binomials 446 kB 642 October 14, 2012
Fractions – Simplify product of binomials and trinomials 817.7 kB 731 October 14, 2012