Pilne! 30pkt wielomiany na szybko 1. Rozłóż wielomian w na czynniki. a) [latex]w(x) = x^{4} - 7 x^{3} [/latex] b) [latex]w(x) = x^{4} - 5 x^{3} + 6 x^{2} [/latex] c) [latex]w(x) = x^{3} + 5 x^{2} + x + 5[/latex] 2. Rozwiąż równanie. a) [latex] x^{

Zadanie 1 [latex]a)w(x)=x^4-7x^3 \w(x)=x^3(x-7)[/latex] ========================= [latex]b)w(x)=x^4-5x^3+6x^2 \w(x)=x^2(x^2-5x+6) \w(x)=x^2(x^2-5x+6,25-6,25+6) \w(x)=x^2[(x-2,5)^2-0,25] \w(x)=x^2(x-2,5-0,5)(x-2,5+0,5) \w(x)=x^2(x-3)(x-2)[/latex] ========================= [latex]c)w(x)=x^3+5x^2+x+5 \w(x)=x^2(x+5)+1(x+5) \w(x)=(x^2+1)(x+5)[/latex] ========================= Zadanie 2 [latex]a)x^3-3x^2-4x+12=0 \x^2(x-3)-4(x-3)=0 \(x^2-4)(x-3)=0 \(x-2)(x+2)(x-3)=0 \x-2=0 quad vee quad x+2=0 quad vee quad x-3=0 \x=2 quad vee quad x=-2 quad vee quad x=3[/latex] [latex]b) frac{1}{1-x^2}+ frac{1}{1+x}=2 \D:x in mathbb{R}setminus{-1;1} \\ frac{1}{(1-x)(1+x)}+ frac{1}{1+x}=2 \ frac{1+1-x}{(1+x)(1-x)} =2quad /cdot (1+x)(1-x) \2-x=2-2x^2 \2x^2-x=0 \x(2x-1)=0 \x=0 quad vee quad 2x-1=0 \x=0 quad vee quad x=0,5[/latex]

1. a)    W(x) = x⁴ - 7x³ = x³(x - 7) b)  Wx = x⁴ - 5x³ + 6x² = x²(x² - 5x + 6) = x²(x² - 3x - 2x + 6) = x²[x(x - 3)-2(x - 3)]= = x²(x - 2)(x - 3) c) W(x) = x³ + 5x² + x + 5 = x²(x + 5) + (x + 5) = (x² + 1)(x  5) 2. a) x³ - 3x² - 4x + 12 = 0 x²(x - 3) - 4(x - 3) = 0 (x² - 4)(x - 3) = 0 (x + 2)(x - 2)(x - 3) = 0 x = -2    v    x = 2    x = 3 x ∈ {-2; 2; 3}  [latex]b)\\frac{1}{1-x^{2}} + frac{1}{1+x}= 2\\frac{1}{(1+x)(1-x)} + frac{1}{1+x} = 2 |*(1+x)(1-x)\\x eq -1\x eq 1\D = R-lbrace{-1; 1 brace}[/latex] [latex]1+1-x = 2(1-x^{2})\\2-x = 2 - 2x^2}\\2x^{2}-x = 0\\x(2x-1) = 0\\x = 0 vee x = frac{1}{2}[/latex]