Algebraické vzorce
Důkaz algebraických vzorců se provádí přímým důkazem:
\( (a+b)^2 = (a + b)(a+b) = (a^2 + ab + b^2 + ab) = a^2 +2ab + b^2 \)
\( (a-b)^2 = (a - b)(a-b) = (a^2 - ab + b^2 - ab) = a^2 - 2ab + b^2 \)
\( (a+b)^3 = (a + b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = a^3 + a^2b + 2a^2b + 2ab^2 + ab^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)
\( (a-b)^3 = (a - b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = a^3 - a^2b - 2a^2b + 2ab^2 + ab^2 - b^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)
\( (a-b)(a+b) = a^2 +ab -ba + b^2 = a^2 - b^2 \)
\( (a+b) (a^2 - ab + b^2) = a^3 -a^2b +ab^2 +a^2b - ab^2 +b^3 = a^3 + b^3 \)
\( (a-b) (a^2 + ab + b^2) = a^3 + a^2b + ab^2 -a^2b - ab^2 - b^3 = a^3 - b^3 \)
| \( (a + b)^2 = a^2 + 2ab + b^2 \) | \( (a - b)^2 = a^2 - 2ab + b^2 \) |
| \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \) | \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \) |
| \( (a^2 + b^2) \Rightarrow \) nelze rozložit |
\( (a^2 - b^2) = (a-b)(a+b) \) |
| \( (a^3 + b^3) = (a+b) (a^2 - ab + b^2) \) | \( (a^3 - b^3) = (a-b) (a^2 + ab + b^2) \) |
\( a = 11x; b = 22y \)
\( (11x+22y)^2 = (11x + 22y)(11x+22y) = ((11x)^2 + 11x22y + (22y)^2 + 11x22y) = (11x)^2 +2\times11x22y + (22y)^2 = 121x^2 + 484xy + 484y^2 \)
\( (11x-22y)^2 = (a - b)(a-b) = (a^2 - ab + b^2 - ab) = a^2 - 2ab + b^2 \)
\( (11x+22y)^3 = (a + b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = a^3 + a^2b + 2a^2b + 2ab^2 + ab^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)
\( (11x-22y)^3 = (a - b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = a^3 - a^2b - 2a^2b + 2ab^2 + ab^2 - b^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)
\( (11x-22y)(11x+22y) = a^2 +ab -ba + b^2 = a^2 - b^2 \)
\( (11x+22y) ((11x)^2 - 11x22y + (22y)^2) = a^3 -a^2b +ab^2 +a^2b - ab^2 +b^3 = a^3 + b^3 \)
\( (11x-22y) ((11x)^2 + 11x22y + (22y)^2) = a^3 + a^2b + ab^2 -a^2b - ab^2 - b^3 = a^3 - b^3 \)