That quickly and effectively to make calculations, simplify mathematical expressions. For this purpose use the mathematical ratios allowing to make expression is shorter, and to simplify calculations.
It is required to you
- - concept of a monomial of a polynomial;
- - formulas of abridged multiplication;
- - actions with fractions;
- - main trigonometrical identities.
1. If in expression there are monomials with identical multipliers, find the sum of coefficients at them and increase by a multiplier, uniform for them. For example, if there is an expression 2·and-4 • and +5 • and + and = (2-4+5+1) a =4a.
2. For simplification of expression use formulas of abridged multiplication. The difference square, the difference of squares, a difference and the sum of cubes belong to the most popular. For example, if there is an expression 256-384+144, present it as 16²-2•16•12+12²= (16-12)²=4²=16.
3. In case expression represents natural fraction, allocate the general multiplier from numerator and a denominator and reduce fraction by it. For example, if it is necessary to reduce fraction (3•a²-6·a· b+3·b²) / (6∙a²-6∙b²), take out from numerator and the general multipliers in numerator it will be 3 denominator, in a denominator 6. Receive expression (3 • (a²-2·a · b+b²)) / (6 ∙ ²-b²)). Reduce numerator and a denominator by 3 and apply to the remained expressions of a formula of abridged multiplication. For numerator it is a difference square, and for a denominator the difference of squares. Receive expression (a-b)² / (2 ∙ (a+b) ∙ (a-b)) having reduced it by the general multiplier of a-b, receive expression (a-b) / (2 ∙ (a+b)) who at concrete values of variables is much easier for considering.
4. If monomials have the identical multipliers built in degree, then at their summation watch that degrees were equal, differently it is impossible to reduce similar. For example, if there is an expression 2∙m²+6•³-m²-4•m³+7, then at data of similar m²+2•m³+7 will turn out.
5. At simplification of trigonometrical identities use formulas for their transformation. Main trigonometrical identity of sin² (x) +cos² (x) =1, sin(x)/cos(x)=tg (x), 1/tg(x) = ctg(x), formulas of the sum and difference of arguments, double, threefold argument and others. For example, (sin(2∙x) - cos(x)) / ctg(x). Paint a formula of a double argument and a cotangent as the cosine relations on a sine. Receive (2 ∙ sin(x) • cos(x) - cos(x)) • sin(x)/cos(x). Take out the general multiplier, cos(x) and reduce fraction of cos (x) • (2 ∙ sin(x) - 1) • sin(x)/cos(x) = (2 ∙ sin(x) - 1) • sin(x).