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Binomial formulas explained simply

The 3 binomial formulas are designed to make life easier for mathematicians. For many students, however, this does not come across as a relief when working with the binomial formulas for the first time. Hopefully this article will provide an enlightenment for anyone interested in this area.

Anyone who is familiar with the calculation of parentheses does not actually need the binomial formulas. Because these inevitably result from the laws of calculation. There is one simple reason why these are still treated in school: they make life easier. The 3 binomial formulas thus represent an "abbreviation". And which student does not like to take the path of least resistance? Before we really get started with the topic, however, you should master the basics of brackets. If you still have doubts, take a look at the following articles. Everyone else can start right away with the first binomial formula.

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First binomial formula

For anyone who can multiply parentheses, the first binomial formula is actually nothing new, even if it seems daunting at first glance. Because this is:

  • 1. Binomial formula: (a + b)2 = a2 + 2ab + b2
  • Derivation: (a + b)2 = (a + b) * (a + b) = a2 + from + ba + b2 = a2 + 2ab + b2


The derivation is interesting for all those who ask themselves: "Where does this actually come from?" Everyone else just needs the math expression I marked in bold. The derivation simply shows how to multiply the brackets (which we have already explained in the section linked above). A few examples demonstrate how to apply the formula:

  • ( 3 + 4 )2 = 32 + 2 · 3 · 4 + 42 = 9 + 24 + 16 = 49
  • ( 1 + 2 )2 =12 + 2 · 1 · 2 + 22 =1 + 4 + 4 = 9


Tip: Take a look at the binomial formula and make it clear what a and b is. And then you substitute the numbers for a and b. If you compare the formula with what I have calculated above, it should become clearer. Our exercises (link at the bottom of the article) should also provide additional enlightenment.

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Second binomial formula

The second binomial formula looks very similar. Only here is there a negative sign. The formula and its derivation follow again:

  • 2. Binomial formula: (a - b)2 = a2 - 2ab + b2
  • Derivation: (a - b)2 = (a - b) * (a - b) = a2 - from -ba + b2 = a2 - 2ab + b2

Here, too, in the end it is a question of seeing "Ok, there is a difference in brackets" and then inserting it. Here, too, two small examples for a better understanding:

  • ( 4 - 2 )2 = 42 -2 · 4 · 2 + (2)2 =16 - 16 + 4 = 4
  • (3 - a)2 = 32 - 2 * 3 * a + a2 = 9 - 6a + a2

Again, the advice: Compare the 2nd binomial formula from above with what was calculated in the examples. Then you should do the exercises that are linked at the bottom of the article.

Third binomial formula

We come to the third - and thus last - binomial formula. This helps multiply two brackets that look like this:

  • 3. Binomial formula: (a + b) (a - b) = a2 - b2
  • Derivation: (a + b) (a - b) = a2 -ab + ba -b2 = a2 - b2

This formula can therefore be used if you have two brackets in which the second variable only behaves differently in terms of the sign. Here, too, some examples will (hopefully) help to clarify:

  • (a + 3) (a - 3) = a2 -32 = a2 - 9
  • (2 + b) (2 - b) = 22 - b2 = 4 - b2

Binomial formulas to the power of 3,4,5 etc., exercises and factoring

In order to find out more about the binomial formulas, there are a number of other articles and offers on this topic below.

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