Margliður – Æfing 1 – Lausnir

1) f(x) = 2x3 – x2 – 43x + 60 og g(x) = 2x – 3

f(x) + g(x) = 2x3 – x2 – 43x + 60 + 2x – 3 = 2x3 – x2 – 41x + 57

f(x) – g(x) = 2x3 – x2 – 43x + 60 – 2x + 3 = 2x3 – x2 – 45x + 63

f(x)·g(x) = (2x3 – x2 – 43x + 60)(2x – 3) = 4x4 – 2x3 – 86x2 + 120x – 6x3 + 3x2 + 129x + 180 =

4x4 – 8x3 – 83x2 + 9x – 180

f(x)/g(x) = x2 + x – 20 þar sem:

2) f(x) = 9x2 + 1 og g(x) = (3x – 1)(3x + 1)

f(x) + g(x) = 9x2 + 1 + 9x2 – 1 = 18x2

f(x) – g(x) = 9x2 + 1 – 9x2 + 1 = 2

f(x)·g(x) = (9x2 + 1)(9x2 – 1) = 81 x2 – 1

f(x)/g(x) = 1 + 2/(9x2 – 1)

3) f(x) = (x – 1)(x2 + x – 1) og g(x) = (x + 1)(x2 + x – 1)

margfalda upp úr svigunum og fæ að:

f(x) = x3 + x2 – x – x2 – x + 1 = x3–2x + 1

g(x) = x3 + x2 – x + x2 + x – 1 = x3 + 2x2– 1

f(x) + g(x) = x3–2x + 1+ x3 + 2x2– 1 = 2x3+ 2x2–2x

f(x) – g(x) = x3–2x + 1- x3 – 2x2+ 1 = – 2x2 – 2x + 2

f(x)·g(x) = (x – 1)(x2 + x – 1)(x + 1)(x2 + x – 1) = (x2 – 1)( (x2 + x – 1)2

= (x2 – 1) (x4 + 2x3 – x2 – 2x -1) = x6 + 2x5 – 2x4 – 6x3 + 2x + 1

f(x)/g(x) = (x – 1)(x2 + x – 1)/ (x + 1)(x2 + x – 1) = (x – 1)/(x + 1) = 1 – 2/(x+1)