Mathematics - one of those sciences that are essential to the existence of mankind.Almost every action, every process associated with the use of mathematics and its basic operations.Many great scientists have made tremendous efforts to ensure that the science to make this easier and more intuitive.Various theorems, axioms and formulas allow students to quickly perceive information and to apply this knowledge in practice.The majority of them remembered lifelong.
most convenient formula that allows students and pupils to cope with the huge examples fractions, rational and irrational expressions are formulas, including abridged multiplication:
1. the sum and difference of cubes:
s3- t3 - the difference;
k3 + l3 - amount.
2. Formula cube sum and difference of the cube:
(f + g) and 3 (h - d) 3;
3. difference of squares:
z2 - v2;
4. squared sum:
(n + m) 2, and so on. D.
Formula sum of the cubes is practically very difficult to memorize and play.This stems from the alternating signs in its decoding.They incorrectly written, confusing with other formulas.
sum of cubes disclosed as follows:
k3 + l3 = (k + l) * (k2 - k * l + l2).
second part of the equation is sometimes confused with a quadratic equation or an expression of the disclosed amount and the square is added to the second term, namely, the «k * l» number 2. However, the amount of formula cubes reveals the only way.Let us prove the equality of the right and left side.
Come reverse, ie, try to show that the second half of the (k + l) * (k2 - k * l + l2) will be equal to the expression k3 + l3.
us open bracket, multiplying terms.For this, first we multiply «k» on each member of the second expression:
k * (k2 - k * l + k2) = k * l2 - k * (k * l) + k * (l2);
then in the same way produce effects with unknown «l»:
l * (k2 - k * l + k2) = l * k2 - l * (k * l) + l * (l2);
simplify the resulting expression of the formula amount of cubes, reveal the braces, and thus give these terms:
(k3 - k2 * l + k * l2) + (l * k2 - l2 * k + l3) = k3 - k2l + kl2+ lk2 - lk2 + l3 = k3 - k2l + k2l + kl2- kl2 + l3 = k3 + l3.
This expression is equal to the initial variant of the sum of the cubes, which is to be shown.
no evidence for expression s3 - t3.This mathematical formula abridged multiplication is called the difference of cubes.She disclosed as follows:
s3 - t3 = (s - t) * (s2 + t * s + t2).
Similarly as in the previous example way prove compliance with the right and left sides.For this reveal brackets multiplying terms:
for an unknown «s»:
s * (s2 + s * t + t2) = (s3 + s2t + st2);
unknown for «t»:
t * (s2 + s * t + t2) = (s2t + st2 + t3);
the transformation and parentheses disclosure of the difference is obtained:
s3 + s2t + st2 - s2t - s2t - t3 = s3 + s2t- s2t - st2 + st2- t3 = s3 - t3 - QED.
To remember which characters are set upon expansion of this expression, it is necessary to pay attention to signs between terms.So, if one is separated from another unknown mathematical symbol "-", then in the first bracket will be negative, and the second - two pluses.If between the cubes is "+" sign, then, accordingly, the first factor will contain a plus and minus of the second, and then a plus.
It can be represented as a small circuit:
s3 - t3 → («negative») * ("plus" "plus");
k3 + l3 → («plus») * ("minus" sign "plus").
Consider this example:
Given the expression (w - 2) 3+ 8. Disclose brackets.
Solution:
(w - 2) 3 + 8 can be expressed as (w - 2) 3 +23
Accordingly, as the sum of the cubes, this expression can be expanded by the formula abridged multiplication:
(w - 2 +2) * ((w - 2) 2 - 2 * (w - 2) + 22);
Then simplify the expression:
w * (w2 - 4w + 4 - 2w + 4 + 4) = w * (w2 - 6w + 12) = w3 - 6w2 + 12w.
Thus, the first part (w - 2) 3 can also be regarded as a cube difference:
(h - d) 3 = h3 - h2 * 3 * 3 + d * h * d2 - d3.
Then, if open it on this formula, you get:
(w - 2) 3 = w3 - 3 * w2 * 2 + 3 * w * 22 - 23 = w3 - 6 * w2 + 12w - 8.
If we add to it a second example of the original, namely, "8", the result is as follows:
(w - 2) 3 + 8 = w3 - w2 * 3 * 3 * 2 + 22 * w - 23 + 8 =w3 - 6 * w2 + 12w.
Thus, we have found a solution to this example in two ways.
important to remember that the key to success in any business, including in solving mathematical examples are perseverance and care.
