Sum of Cubes and Their Difference: Abbreviated Multiplication Formulas

Mathematics is one of those sciences, without which the existence of humanity is impossible. Almost every action, every process is associated with the use of mathematics and its elementary actions. Many great scientists have made great efforts to make this science easier and more understandable. Various theorems, axioms and formulas allow students to quickly perceive information and apply knowledge in practice. However, most of them are remembered throughout life.

The most convenient formulas that allow students and schoolchildren to cope with gigantic examples, fractions, rational and irrational expressions are formulas, including abbreviated multiplication:

1. sums and differences of cubes :

s ³ - t ³ is the difference;

k ³ + l ³ is the sum.

2. the formula of the cube of the sum, as well as the cube of difference:

(f + g) ³ and (h - d) ^3;

3. square difference:

z ² is v ² ;

4. squared amount:

(n + m) ² , etc.

The formula for the sum of cubes is almost the most difficult to remember and play. The reason for this is the alternating signs in its decoding. They are spelled incorrectly, confused with other formulas.

The sum of the cubes is disclosed as follows:

k ³ + l ³ = (k + l) * (k ² - k * l + l ² ).

The second part of the equation is sometimes confused with the quadratic equation or the open expression of the square of the sum and added to the second term, namely, to "k * l" is number 2. However, the formula for the sum of cubes is revealed only in this way. Let's prove the equality of the right and left sides.

Let's go from the opposite, that is, try to show that the second half (k + l) * (k ² - k * l + l ² ) will be equal to the expression k ³ + l ³ .

We open the brackets by multiplying the terms. To do this, first multiply “k” by each member of the second expression:

k * (k ² - k * l + k ² ) = k * l ² - k * (k * l) + k * (l ² );

then in the same way we perform an action with an unknown “l”:

l * (k ² - k * l + k ² ) = l * k ² - l * (k * l) + l * (l ² );

we simplify the resulting expression of the formula, the sum of cubes, open the brackets, and at the same time give similar terms:

(k ³ - k ² * l + k * l ² ) + (l * k ² - l ² * k + l ³ ) = k ³ - k ² l + kl ² + lk ² - lk ² + l ³ = k ³ - k ² l + k ² l + kl ² - kl ² + l ³ = k ³ + l ³ .

This expression is equal to the original version of the formula, the sum of the cubes, and this was required to be shown.

Find the proof for the expression s ³ - t ³ . This mathematical formula of abbreviated multiplication is called the difference of cubes. It is disclosed as follows:

s ³ - t ³ = (s - t) * (s ² + t * s + t ² ).

In the same way as in the previous example, we prove the correspondence of the right and left parts. To do this, open the brackets, multiplying the terms:

for unknown "s":

s * (s ² + s * t + t ² ) = (s ³ + s ² t + st ² );

for the unknown "t":

t * (s ² + s * t + t ² ) = (s ² t + st ² + t ³ );

when converting and expanding the brackets of this difference, it turns out:

s ³ + s ² t + st ² - s ² t - s ² t - t ³ = s ³ + s ² t– s ² t - st ² + st ² - t ³ = s ³ - t ³ - as required to prove.

In order to remember what signs are placed when revealing such an expression, it is necessary to pay attention to the signs between the terms. So, if one unknown is separated from the other by the mathematical symbol “-”, then in the first bracket there will be a minus, and the second - two pluses. If the “+” sign is located between the cubes, then, accordingly, the first factor will contain plus, and the second minus, and then plus.

This can be represented in the form of a small diagram:

s ³ - t ³ → ("minus") * ("plus" "plus");

k ³ + l ³ → (“plus”) * (“minus” “plus”).

Consider an example:

The expression (w - 2) ³ + 8 is given. It is necessary to open the brackets.

Decision:

(w - 2) ³ + 8 can be represented as (w - 2) ³ + 2 ³

Accordingly, as the sum of cubes, this expression can be decomposed according to the formula of abbreviated multiplication:

(w - 2 + 2) * ((w - 2) ² - 2 * (w - 2) + 2 ² );

Then simplify the expression:

w * (w ² - 4w + 4 - 2w + 4 + 4) = w * (w ² - 6w + 12) = w ³ - 6w ² + 12w.

Moreover, the first part of (w - 2) ³ can also be considered as a cube of difference:

(h - d) ³ = h ³ - 3 * h ² * d + 3 * h * d ² - d ³ .

Then, if you open it according to this formula, you get:

(w - 2) ³ = w ³ - 3 * w ² * 2 + 3 * w * 2 ² - 2 ³ = w ³ - 6 * w ² + 12w - 8.

If we add to it the second part of the original example, namely “+8”, the result will be as follows:

(w - 2) ³ + 8 = w ³ - 3 * w ² * 2 + 3 * w * 2 ² - 2 ³ + 8 = w ³ - 6 * w ² + 12w.

Thus, we found a solution to this example in two ways.

It must be remembered that the key to success in any business, including solving mathematical examples, is perseverance and attentiveness.