It is rather simple to solve identities. For this purpose it is required to make identical transformations until the goal is achieved. Thus, by means of the simplest arithmetic actions the objective will be solved.
It is required to you
- - paper;
- - handle.
Instruction
1. The simplest example of such transformations – algebraic formulas of abridged multiplication (such as square of the sum (difference), difference of squares, sum (difference) of cubes, cube of the sum (difference)). Besides there is a set of logarithmic and trigonometrical formulas which in essence are the same identities.
2. Really, the square of the sum of two composed is equal to a square of the first plus the doubled work by the first on the second and plus a square of the second, that is (a+b) ^2 = (a+b) (a+b) =a^2+ab +ba+b^2=a^2+2ab+b^2. Simplify expression (a-b) ^2 +4ab. (a-b) ^2 +4ab = a^2-2ab+b^2 +4ab=a^2+2ab+b^2=(a+b) ^2. At the higher mathematical school if to understand, identical transformations – the foremost of the foremost. But there they are considered something self-evident. Their purpose not always expression simplification, and sometimes and complication, on purpose, as it was already told, achievements of a goal. Any proper rational fraction can be presented in the form of the sum of final number of protozoa drobeypm (x)/Qn (x) = by A1 / (x-a) + A2 / (x-a) ^2+ … +Ak / (x-a) ^k+ …+ (M1x+N1)/(x^2+2px+q) + … + (M2x+N2)/(x^2+2px+q) ^s.
3. Example. Identical transformations to spread out to the simplest fractions (x^2)/(1-x^4). Spread out expression 1-х^4=(1-x) (1+x) (x^2+1). (x^2)/(1-x^4) = A / (1-x) + B / (x+1) + (Cx+D)/(x^2+1) Reduce the sum to a common denominator and equate numerators of fractions in both parts of equality.X^2=A (x+1) (x^2+1) + B(1-x)(x^2+1)+ (Cx+D) (1-x^2) Notice that: At x = 1: 1 = 4A, A = 1/4; At x = - 1: 1 = 4B, B = 1/4. Coefficients at x^3: A-B-C=0, from where With =0koeffitsiyenty at x^2: A+B-D=1 and D=-1/2itak, (x^2)/(1-x^4) =1 / (1-x) + 1 / (4 (x+1)) – 1 / (2 (x^2+1)).