a) [latex] W(x) = 4x^4 - 49x^2 - 56x - 16\ W(x) = (2x^2 - 7x)(2x^2 + 7x) - 4(7x + 7x + 4)\ W(x) = (2x^2 - 7x)(2x^2 + 7x) - 4(2x^2 + 7x - 2x^2 + 7x + 4)\ W(x) = (2x^2 - 7x)(2x^2 + 7x) - 4(2x^2 + 7x) + 4(2x^2 -7x -4)\ W(x) = (2x^2 - 7x -4)(2x^2 + 7x)+ 4(2x^2 -7x -4)\ W(x) = (2x^2 - 7x -4)(2x^2 + 7x + 4)\ W(x) = (2(x-4)(x+frac{1}{2}))(2x^2 + 7x + 4)\ W(x) = (2(x-4)(x+frac{1}{2}))(2(x + frac{7 + sqrt{17}}{4})(x + frac{7 - sqrt{17}}{4}))\ W(x) = 4(x-4)(x+frac{1}{2})(x + frac{7 + sqrt{17}}{4})(x + frac{7 - sqrt{17}}{4})[/latex] b) W(x)= 0 , gdy x= 4 lub x=-0,5 lub [latex]x = frac{-7 + sqrt{17}}{4}[/latex] lub [latex]x= frac{-7 - sqrt{17}}{4}[/latex] c) [latex]x = frac{-7 + sqrt{17}}{4}[/latex] lub [latex]x= frac{-7 - sqrt{17}}{4}[/latex]
[latex]W(x)=4x^4-49x^2-56x -16[/latex] a) [latex]4x^4-49x^2-56x -16 = 4x^4 + 14x^3 + 8x^2 - 14x^3 - 49x^2 - 28x - 8x^2 - 28x - 16 =[/latex] [latex]= 2x^2 cdot (2x^2+7x+4) - 7x cdot (2x^2+7x +4) - 4cdot (2x^2 +7x +4) =[/latex] [latex]= (2x^2-7x -4) cdot (2x^2+7x+4)=(x -4) cdot (2x+1) cdot (2x +frac{7 + sqrt{17}}{2})cdot (x +frac{7 -sqrt{17}}{4}) [/latex] b) [latex]4x^4-49x^2-56x -16 = 0[/latex] [latex](x -4) cdot (2x+1) cdot (2x +frac{7 + sqrt{17}}{2})cdot (x +frac{7 -sqrt{17}}{4}) = 0[/latex] [latex]x -4 = 0 vee 2x+1 = 0 vee 2x +frac{7 + sqrt{17}}{2} = 0 vee x +frac{7 -sqrt{17}}{4} = 0[/latex] [latex]x -4 = 0 [/latex] [latex]x_1 = 4[/latex] [latex]2x+1 = 0[/latex] [latex]2x = - 1 / : 2[/latex] [latex]x_2 = - frac{1}{2}[/latex] [latex]2x +frac{7 + sqrt{17}}{2} = 0 [/latex] [latex]2x = - frac{7 + sqrt{17}}{2} / : 2[/latex] [latex]x_3 = frac{- 7 - sqrt{17}}{4}[/latex] [latex]x +frac{7 -sqrt{17}}{4} = 0 [/latex] [latex]x = - frac{7 -sqrt{17}}{4}[/latex] [latex]x_4 = frac{sqrt{17} - 7}{4}[/latex] c) Niewymierne pierwiastki równania W(x) = 0 to [latex]x_3 = frac{- 7 - sqrt{17}}{4}[/latex] [latex]x_4 = frac{sqrt{17} - 7}{4} [/latex] ****************************************** Obliczenia dodatkowe: Postać iloczynowa trójmianu kwadratowego: ax² + bx + c = a·(x - x₁)·(x - x₂) [latex]2x^2-7x -4[/latex] [latex]Delta = 49 + 32 = 81; sqrt{Delta} = 9[/latex] [latex]x_1 = frac{7 + 9}{4} = frac{16}{4} = 4[/latex] [latex]x_2 = frac{7 - 9}{4} = frac{-2}{4} = -frac{1}{2}[/latex] [latex]2x^2-7x -4 =2 cdot (x -4) cdot (x+frac{1}{2}) = (x -4) cdot (2x+1) [/latex] [latex]2x^2+7x+4 [/latex] [latex]Delta = 49 - 32 = 17; sqrt{Delta} = sqrt{17}[/latex] [latex]x_1 = frac{-7- sqrt{17}}{4}[/latex] [latex]x_2 = frac{-7+ sqrt{17}}{4}[/latex] [latex]2x^2+7x+4 = 2 cdot (x +frac{7 + sqrt{17}}{4})cdot (x +frac{7 -sqrt{17}}{4}) = (2x +frac{7 + sqrt{17}}{2})cdot (x +frac{7 -sqrt{17}}{4}) [/latex]
