Odpowiedź:
1. [tex]\frac{x^{2} -9}{x^{2} -3x+2}[/tex] × [tex]\frac{x^{2} -2x}{x^{2} +2x-3}[/tex] × [tex]\frac{x^{2} -x}{x^{2} }[/tex]
a) Założenia => Dziedzina:
1) [tex]x^{2} -3x + 2[/tex] ≠ 0
Δ = 9 - 4(1)(2) = 9 - 8 = 1
[tex]\sqrt{delta}[/tex] = 1
x1 = [tex]\frac{3-1}{2}[/tex] = 1
x2 = [tex]\frac{3+1}{2}[/tex] = 2
x ≠ 1 ∧ x ≠ 2
2) [tex]x^{2} +2x - 3[/tex] ≠ 0
Δ = 4 - 4(1)(-3) = 4 + 12 = 16
[tex]\sqrt{delta}[/tex] = 4
x1 = [tex]\frac{-2-4}{2}[/tex] = -3
x2 = [tex]\frac{-2+4}{2}[/tex] = 1
x ≠ -3 ∧ x ≠ 1
3) [tex]x^{2}[/tex] ≠ 0
x ≠ 0
D: x∈R - {-3, 0, 1, 2}
b) Obliczenia:
[tex]\frac{x^{2} -9}{x^{2} -3x+2}[/tex] × [tex]\frac{x^{2} -2x}{x^{2} +2x-3}[/tex] × [tex]\frac{x^{2} -x}{x^{2} }[/tex] =
= [tex]\frac{(x-3)^{2} }{(x-1)(x-2)}[/tex] × [tex]\frac{x(x-2)}{(x+3)(x-1)}[/tex] × [tex]\frac{x(x-1)}{x^{2}}[/tex] =
= [tex]\frac{(x-3)^{2} }{(x-1)(x-2)}[/tex] × [tex]\frac{x(x-2)}{(x+3)(x-1)}[/tex] × [tex]\frac{x-1}{x}[/tex] =
= [tex]\frac{(x-3)^{2}x(x-2)}{(x-1)^{2} (x-2)(x+3)}[/tex] × [tex]\frac{x-1}{x}[/tex] =
= [tex]\frac{(x-3)^{2}x(x-2)(x-1)}{x(x-1)^{2} (x-2)(x+3)}[/tex] =
= [tex]\frac{x-3}{x-1}[/tex]
2. [tex]\frac{x^{2} +x-6}{x+5}[/tex] : [tex]\frac{x-2}{x^{2} +2x-15}[/tex]
a) Założenia => Dziedzina:
1) x + 5 ≠ 0
x ≠ -5
2) x² + 2x - 15 ≠ 0
Δ = 4 - 4(1)(-15) = 4 + 60 = 64
[tex]\sqrt{delta}[/tex] = 8
x1 = [tex]\frac{-2-8}{2}[/tex] = -5
x2 = [tex]\frac{-2+8}{2}[/tex] = 3
x ≠ 3 ∧ x ≠ -5
D: x∈R - {-5, 3}
b) Obliczenia:
[tex]x^{2} +x-6[/tex] = 0
Δ = 1 - 4(1)(-6) = 1 + 24 = 25
[tex]\sqrt{delta}[/tex] = 5
x1 = [tex]\frac{-1-5}{2}[/tex] = -3
x2 = [tex]\frac{-1+5}{2}[/tex] = 2
[tex]x^{2} +x-6[/tex] = (x + 3)(x - 2)
[tex]\frac{x^{2} +x-6}{x+5}[/tex] : [tex]\frac{x-2}{x^{2} +2x-15}[/tex] =
= [tex]\frac{x^{2} +x-6}{x+5}[/tex] : [tex]\frac{x-2}{(x-3)(x+5)}[/tex] =
= [tex]\frac{(x+3)(x-2)}{(x+5)}[/tex] × [tex]\frac{(x-3)(x+5)}{(x-2)}[/tex] = (D: x∈R - {-5, 2, 3})
= [tex]\frac{(x + 3)(x-2)(x-3)(x+5)}{(x+5)(x-2)}[/tex] =
= (x + 3) × (x - 3) = x² - 9
