What is 3 3 4 5

Binomial formulas to the power of 3, 4 and 5

The "classic" three binomial formulas are each based on a squared term, that is, on a term that is taken to the power of 2. The question arises, of course, whether it is too binomial formulas in case the The exponent of the binomial is greater than two is. In fact, there are binomial formulas for these rare cases as well. The reason why these are rather unknown is that the expressions are significantly more complicated and not as easy to learn as those of the binomial formulas to the power of 2.

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In the following we will calculate three possible cases of higher exponents using binomial formulas. As an example, we use the first binomial formula as a guide, i.e. a sum in brackets.

1. Binomial formula: $ (a \ textcolor {red} {+} b) ^ 2 = a ^ 2 + 2 \ cdot a \ cdot b + b ^ 2 $

Binomial formulas with the exponent 3

To simplify binomial terms with the exponent $ 3 $, we first solve the power. In doing so, we split the high 3 term into a multiplication from a single bracket and a high 2 term, which we can in turn solve with the binomial formulas we know.

$ (a + b) ^ 3 = (a + b) ^ 2 \ times (a + b) = (a ^ 2 + 2 \ times a \ times b + b ^ 2) \ times (a + b) $

Now we have to multiply the two remaining brackets, that is, we take each number of one bracket with that of the other times and connect them with a plus sign. This results in a very complicated expression at first.

$ (a + b) ^ 3 = (a \ times a ^ 2) + (a \ times 2 \ times a \ times b) + (a \ times b ^ 2) + (b \ times a ^ 2) + ( b \ cdot 2 \ cdot a \ cdot b) + (b \ cdot b ^ 2) $

If we add up all the multiplications as far as possible, we get the following expression:

$ (a + b) ^ 3 = a ^ 3 + \ textcolor {red} {(2 \ cdot a ^ 2 \ cdot b)} + \ textcolor {blue} {(a \ cdot b ^ 2)} + \ textcolor {red} {(b \ cdot a ^ 2)} + \ textcolor {blue} {(2 \ cdot a \ cdot b ^ 2)} + b ^ 3 $

The terms marked in color can be summarized:

$ (a + b) ^ 3 = a ^ 3 + \ textcolor {red} {3 \ cdot a ^ 2 \ cdot b} + \ textcolor {blue} {3 \ cdot a \ cdot b ^ 2} + b ^ 3 $

This formula can be formulated accordingly for the case of a difference.

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The binomial formulas with the exponent $ 3 $

$ (a + b) ^ 3 = a ^ 3 + 3 \ times a ^ 2 \ times b + 3 \ times a \ times b ^ 2 + b ^ 3 $
$ (a-b) ^ 3 = a ^ 3 - 3 \ times a ^ 2 \ times b + 3 \ times a \ times b ^ 2 - b ^ 3 $
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$ (x + 2) ^ 3 = x ^ 3 + 3 \ times x ^ 2 \ times 2 + 3 \ times \ times \ times 4 + 2 ^ 3 $

$ (x + 2) ^ 3 = x ^ 3 + 6 \ times x ^ 2 + 12 \ times x + 8 $