Wyznacz sumę f+g oraz różnicę...- Zadanie 1: MATeMAtyka 2. Zakres podstawowy i rozszerzony

Rozwiązanie

$a) f (x) + g (x) = \frac{3}{4} x^{6} - 2 x^{4} + \frac{5}{4} x^{2} + 4 + \frac{1}{4} x^{6} + 2 x^{4} + x^{3} - \frac{1}{4} x^{2} - 2 x + 2 = x^{6} + x^{3} + x^{2} - 2 x + 6$

$f (x) - g (x) = (\frac{3}{4} x^{6} - 2 x^{4} + \frac{5}{4} x^{2} + 4) - (\frac{1}{4} x^{6} + 2 x^{4} + x^{3} - \frac{1}{4} x^{2} - 2 x + 2) = \frac{3}{4} x^{6} - 2 x^{4} + \frac{5}{4} x^{2} + 4 - \frac{1}{4} x^{6} - 2 x^{4} - x^{3} + \frac{1}{4} x^{2} + 2 x - 2 = \frac{1}{2} x^{6} - 4 x^{4} - x^{3} + \frac{3}{2} x^{2} + 2 x + 2$

$b) f (x) + g (x) = - 2 x^{2} + 4 x - x^{6} + 2 - 5 x + 3 x^{2} + x^{6} - 3 x^{5} = - 3 x^{5} + x^{2} - x + 2$

$f (x) - g (x) = (- 2 x^{2} + 4 x - x^{6} + 2) - (- 5 x + 3 x^{2} + x^{6} - 3 x^{5}) = - 2 x^{2} + 4 x - x^{6} + 2 + 5 x - 3 x^{2} - x^{6} + 3 x^{5} = - 2 x^{6} + 3 x^{5} - 5 x^{2} + 9 x + 2$

$c) f (x) + g (x) = 3 x^{4} + 2 x^{7} - 5 + 4 x - 3 x + x^{5} - 2 x^{7} + 2 = x^{5} + 3 x^{4} + x - 3$

$f (x) - g (x) = (3 x^{4} + 2 x^{7} - 5 + 4 x) - (- 3 x + x^{5} - 2 x^{7} + 2) = 3 x^{4} + 2 x^{7} - 5 + 4 x + 3 x - x^{5} + 2 x^{7} - 2 = 4 x^{7} - x^{5} + 3 x^{4} + 7 x - 7$