Mathematics is one of those sciences without whichthe existence of mankind is impossible. Almost every action, every process involves the use of mathematics and its elementary actions. Many great scientists have made great efforts to make this science easier and more understandable. Different theorems, axioms and formulas allow students to more quickly perceive information and apply knowledge in practice. However, most of them are remembered throughout their lives.
The most convenient formulas for studentsand schoolchildren cope with giant examples, fractions, rational and irrational expressions, are formulas, including shortened multiplication:
1. sums and differences of cubes:
from3 - t3 - difference;
to3 + l3 - amount.
2. The formula of the cube of the sum, as well as the cube of the difference:
(f + g)3 and (h-d)3;
3. difference of squares:
z2 - at2;
4. The square of the sum:
(n + m)2 etc.
The formula of the sum of cubes is almost the most difficult to remember and reproduce. The reason for this is the alternating signs in its decoding. They are incorrectly written, confusing with other formulas.
The sum of the cubes is expanded as follows:
to3 + l3 = (k + l) * (k2 - k * l + l2).
The second part of the equation is sometimes confused withthe quadratic equation or the expanded expression of the square of the sum and add to the second term, namely to "k * l" the number 2. However, the formula the sum of the cubes is disclosed only so. Let us prove the equality of the right and left parts.
Let's go from the opposite, that is, we will try to show that the second half (k + l) * (k2 - k * l + l2) will be equal to the expression k3 + l3.
We open the brackets, multiplying the summands. To do this, first multiply "k" by each term of the second expression:
k * (k2 - k * l + k2) = k * l2 - k * (k * l) + k * (l2);
then in the same way we perform an action with an unknown "l":
l * (k2 - k * l + k2) = l * k2 - l * (k * l) + l * (l2);
we simplify the resulting expression of the formula the sum of the cubes, open the brackets, and at the same time, we give similar terms:
(to3 - to2* l + k * l2) + (l * k2 - l2* k + l3) = k3 - to2l + kl2 + lk2 - lk2 + l3 = k3 - to2l + k2l + kl2- kl2 + l3 = k3 + l3.
This expression is equal to the original version of the formula the sum of the cubes, and this is what we wanted to show.
We find the proof for the expression s3 - t3. This mathematical formula of reduced multiplication is called the difference of cubes. It is disclosed as follows:
from3 - t3 = (s - t) * (s2 + t * s + t2).
Similarly, as in the previous example, we prove the correspondence between the right and left parts. To do this, we expand the brackets, multiplying the terms:
for the unknown "s":
s * (s2 + s * t + t2) = (s3 + s2t + st2);
for the unknown "t":
t * (s2 + s * t + t2) = (s2t + st2 + t3);
when converting and expanding the parentheses of a given difference, we get:
from3 + s2t + st2 - from2t - s2t - t3 = s3 + s2t- s2t-st2+ st2- t3= s3 - t3 - which was to be proved.
In order to remember which signs are putwhen disclosing 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 there is a "+" sign between the cubes, then, respectively, the first multiplier will contain a plus, and the second minus, and then a plus.
This can be represented in the form of a small scheme:
from3 - t3 → ("minus") * ("plus" "plus");
to3 + l3 → ("plus") * ("minus" "plus").
Let's consider an example:
Given the expression (w - 2)3 + 8. It is necessary to open the brackets.
Decision:
(w-2)3 + 8 can be represented in the form (w - 2)3 + 23
Accordingly, as a sum of cubes, this expression can be decomposed according to the formula of abridged multiplication:
(w-2 + 2) * ((w-2)2 - 2 * (w-2) + 22);
Then we simplify the expression:
w * (w2 - 4w + 4 - 2w + 4 + 4) = w * (w2 - 6w + 12) = w3 - 6w2 + 12w.
In this case, the first part (w-2)3 can also be considered as a cube of difference:
(h-d)3 = h3 - 3 * h2* d + 3 * h * d2 - d3.
Then, if you open it using this formula, you get:
(w-2)3 = w3 - 3 * w2 * 2 + 3 * w * 22 - 23 = w3 - 6 * w2 + 12w - 8.
If you add to it the second part of the original example, namely "+8", the result is as follows:
(w-2)3 + 8 = w3 - 3 * w2 * 2 + 3 * w * 22 - 23 + 8 = w3 - 6 * w2 + 12w.
Thus, we have found the solution of this example in two ways.
It is necessary to remember that diligence and attentiveness are the key to success in any business, including in solving mathematical examples.
