Wielomiany 1. Wyznacz iloczyn pierwiastków równania: (x²-3)(x+2)(3x+2)=0 2. Rozwiąż równania: a) (x³+4x)(x³-3x)=0 b) 3x³-2x²-x=0 3. Dana jest suma algebraiczna w(x)=-x²-3x+1. Oblicz wartość tej sumy dla: a) x=[latex]-1 frac{2}{3} [/latex] b) x=

1. (x²-3)(x+2)(3x+2)=0 x²-3=0  x²=3 x=√3 x+2=0 x=-2 3x+2=0 3x=-2  /:3 x= -2/3 √3 * (-2) * (-2/3)= [latex] frac{4}{3} sqrt{3} [/latex] 2. a) (x³+4x)(x³-3x)=0 x(x²+4)x(x²-3)=0 x²(x²+4)(x-√3)(x+√3)=0 x²=0 ∨ x²+4=0 ∨ x-√3=0 ∨ x+√3=0 x=0 ∨ x²=-4    ∨    x=√3    ∨ x=-√3 x=0 ∨ x=√3 ∨ x=-√3 to jest rozwiazanie x²=-4   zadna liczba nie spelnia tego rownania, dlatego to ominelismy b) 3x³-2x²-x=0 x(3x²-2x-1)=0 x=0 ∨ 3x²-2x-1=0 Δ=4-4*3*(-1)=4+12=16 √Δ=4 x₁=[latex] frac{2-4}{6} = frac{-2}{6} = -frac{1}{3} [/latex] x₂=[latex] frac{2+4}{6} =1[/latex] x=0 ∨ x=-1/3  ∨ x=1 3.  w(x)=-x²-3x+1 a) x=[latex] -1frac{2}{3} [/latex] [latex]-(-1 frac{2}{3} )^{2} -3*(-1 frac{2}{3} )+1=-(- frac{5}{3} )^{2} -3*(- frac{5}{3})+1= [/latex]  [latex] - frac{25}{9} +5+1=6- frac{25}{9} = frac{54}{9} - frac{25}{9} = frac{29}{9}=3 frac{2}{9} [/latex] b) x=2-3√2 -(2-3√2)²-3(2-3√2)+1=-(4-12√2+18)-6+9√2+1=-4+12√2-18-6+9√2+1=21√2-22-6+1=21√2-27 4. a) 2(3x-2)²-(3-4x)²=2(9x²-12x+4)-(9-24x+16x²)=18x²-24x+8-9+24x-16x²= 2x²-1 b) 2(x-3)²-[latex] frac{1}{2} [/latex] (2x-6)x=2(x²-6x+9)-[latex] frac{1}{2} [/latex] (2x²-6x)=2x²-12x+18-x²+3x=x²-9x+18 c) (2x²+3x)(4x-2)-(3x-2)²=8x³-4x²+12x²-6x-(9x²-12x+4)=8x³+8x²-6x-9x²+12x-4=8x³-x²+6x-4 d) (3x-2)(x-6)(x-2)-(2x-6)²=(3x-2)(x²-2x-6x+12)-(4x²-24x+36)=(3x-2)(x²-8x+12)-4x²+24x-36=3x³-24x²+36-2x²+16x-24-4x²+24x-36=3x³ -30x²+40x-24 5.  (-1)³+k*(-1)²+p*(-1)+5=14 -1+k-p=14 k-p=15 2³+k*2²+p*2+5=17 8+4k+2p+5=17 4k+2p=17-8-5 4k+2p=4   /:2 2k+p=2 k-p=15 2k+p=2 3k=17  /:3 k=17/3=5 i 2/3 k-p=15 p=k-15 p=17/3  - 15= 17/3  -  45/3= -28/3= -9 i 1/3 k=5 i 2/3 p= -9 i 1/3 6. w(-2)=2*(-2)³-(-2)²+3m*(-2)+6=0 2*(-8)-4-6m+6=0 -16-4-6m+6=0 -20-6m+6=0 -14-6m=0 -14=6m  /:6 m=-14/6= -2 i 1/3 7.  w(x)=2x³-x²-(3m-4)x+9 w(-1)=2*(-1)³-(-1)²-(3m-4)*(-1)+9=0 -2-1-(-3m+4)+9=0 -3+3m-4+9=0 3m+2=0 3m=-2  /:3 m= -2/3 8.  2a+2b=60 /:2 a+b=30 b=30-a ab=144 a(30-a)=144 30a-a²-144=0 -a²+30a-144=0 zalozenia: a>0 i b>0 Δ=900-4*(-1)*(-144)=900-576=324 √Δ=18 a₁=[latex] frac{-30-18}{-2} = frac{-48}{-2} =24[/latex] a₂=[latex] frac{-30+18}{-2} = frac{-12}{-2} =6[/latex] a+b=30 24+b=30 b=30-24 b=6 a+b=30 6+b=30 b=30-6 b=24 boki maja dlugosc 6cm i 24cm 9. x²+(x+2)²+(x+4)²=83 x²+x²+4x+4+x²+8x+16=83 3x²+12x+20=83 3x²+12x+20-83=0 3x²+12x-63=0 Δ=144-4*3*(-63)=144+756=900 √Δ=30 x₁=[latex] frac{-12+30}{6} = frac{18}{6} =3[/latex] x₂=[latex] frac{-12-30}{6} = frac{-42}{6} =-7[/latex] 3,5,7 lub -7,-5,-3