Господин Экзамен
x^4-x^3-x^2-2*x+1 если x=-1/4
Решение
Вы ввели
[src]
4 3 2 x - x - x - 2*x + 1
$$x^{4} - x^{3} - x^{2} - 2 x + 1$$
x^4 - x^3 - x^2 - 2*x + 1
Разложение на множители
[src]
/ _____________________________________________________________________________________________________________________________\ / _____________________________________________________________________________________________________________________________\ / _____________________________________________________________________________________________________________________________ \ / _____________________________________________________________________________________________________________________________\ | / _______________ | | / _______________ | | / _______________ | | / _______________ | | / / _____ | | / / _____ | | / / _____ | | / / _____ | | / 11 / 223 \/ 597 7 21 | | / 11 / 223 \/ 597 21 7 | | / 11 / 223 \/ 597 21 7 | | / 11 / 223 \/ 597 7 21 | | / -- - 2*3 / --- + ------- - ----------------------- + ------------------------------------------------------------------- | | / -- - 2*3 / --- + ------- - ------------------------------------------------------------------- - ----------------------- | | / -- - 2*3 / --- + ------- - ------------------------------------------------------------------- - ----------------------- | | / -- - 2*3 / --- + ------- - ----------------------- + ------------------------------------------------------------------- | | _______________________________________________________ / 6 \/ 432 48 _______________ _______________________________________________________ | | _______________________________________________________ / 6 \/ 432 48 _______________________________________________________ _______________ | | / 6 \/ 432 48 _______________________________________________________ _______________ _______________________________________________________| | _______________________________________________________ / 6 \/ 432 48 _______________ _______________________________________________________ | | / _______________ / / _____ / _______________ | | / _______________ / / _______________ / _____ | | / / _______________ / _____ / _______________ | | / _______________ / / _____ / _______________ | | / / _____ / / 223 \/ 597 / / _____ | | / / _____ / / / _____ / 223 \/ 597 | | / / / _____ / 223 \/ 597 / / _____ | | / / _____ / / 223 \/ 597 / / _____ | | / 11 / 223 \/ 597 7 / 18*3 / --- + ------- / 11 / 223 \/ 597 7 | | / 11 / 223 \/ 597 7 / / 11 / 223 \/ 597 7 18*3 / --- + ------- | | / / 11 / 223 \/ 597 7 18*3 / --- + ------- / 11 / 223 \/ 597 7 | | / 11 / 223 \/ 597 7 / 18*3 / --- + ------- / 11 / 223 \/ 597 7 | | / -- + 2*3 / --- + ------- + ----------------------- / \/ 432 48 4* / -- + 2*3 / --- + ------- + ----------------------- | | / -- + 2*3 / --- + ------- + ----------------------- / 4* / -- + 2*3 / --- + ------- + ----------------------- \/ 432 48 | | / 4* / -- + 2*3 / --- + ------- + ----------------------- \/ 432 48 / -- + 2*3 / --- + ------- + ----------------------- | | / -- + 2*3 / --- + ------- + ----------------------- / \/ 432 48 4* / -- + 2*3 / --- + ------- + ----------------------- | | / 12 \/ 432 48 _______________ / / 12 \/ 432 48 _______________ | | / 12 \/ 432 48 _______________ / / 12 \/ 432 48 _______________ | | / / 12 \/ 432 48 _______________ / 12 \/ 432 48 _______________ | | / 12 \/ 432 48 _______________ / / 12 \/ 432 48 _______________ | | / / _____ / / / _____ | | / / _____ / / / _____ | | / / / _____ / / _____ | | / / _____ / / / _____ | | / / 223 \/ 597 / / / 223 \/ 597 | | / / 223 \/ 597 / / / 223 \/ 597 | | / / / 223 \/ 597 / / 223 \/ 597 | | / / 223 \/ 597 / / / 223 \/ 597 | | / 18*3 / --- + ------- / / 18*3 / --- + ------- | | / 18*3 / --- + ------- / / 18*3 / --- + ------- | | / / 18*3 / --- + ------- / 18*3 / --- + ------- | | / 18*3 / --- + ------- / / 18*3 / --- + ------- | | 1 \/ \/ 432 48 \/ \/ \/ 432 48 | | 1 \/ \/ 432 48 \/ \/ \/ 432 48 | | 1 \/ \/ \/ 432 48 \/ \/ 432 48 | | 1 \/ \/ 432 48 \/ \/ \/ 432 48 | 1*|x + - - - ----------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------|*|x + - - + ----------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------|*|x + - - - -------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------|*|x + - - - ----------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------| \ 4 2 2 / \ 4 2 2 / \ 4 2 2 / \ 4 2 2 /
$$\left(x + \left(- \frac{1}{4} + \frac{\sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}}{2} + \frac{\sqrt{- \frac{21}{4 \sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}} - 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{6}}}{2}\right)\right) 1 \left(x - \left(\frac{1}{4} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{6} + \frac{21}{4 \sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}}}}{2} + \frac{\sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}}{2}\right)\right) \left(x - \left(- \frac{\sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}}{2} + \frac{1}{4} + \frac{\sqrt{- \frac{21}{4 \sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}} - 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{6}}}{2}\right)\right) \left(x - \left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{6} + \frac{21}{4 \sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}}}}{2} + \frac{1}{4} + \frac{\sqrt{\frac{7}{18 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}} + \frac{11}{12} + 2 \sqrt[3]{\frac{\sqrt{597}}{48} + \frac{223}{432}}}}{2}\right)\right)$$
(((1*(x - (1/4 - sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))/2 - sqrt(11/6 - 2*(223/432 + sqrt(597)/48)^(1/3) - 7/(18*(223/432 + sqrt(597)/48)^(1/3)) + 21/(4*sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))))/2)))*(x - (1/4 + sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))/2 + sqrt(11/6 - 2*(223/432 + sqrt(597)/48)^(1/3) - 21/(4*sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))) - 7/(18*(223/432 + sqrt(597)/48)^(1/3)))/2)))*(x - (1/4 - sqrt(11/6 - 2*(223/432 + sqrt(597)/48)^(1/3) - 21/(4*sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))) - 7/(18*(223/432 + sqrt(597)/48)^(1/3)))/2 + sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))/2)))*(x - (1/4 - sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))/2 + sqrt(11/6 - 2*(223/432 + sqrt(597)/48)^(1/3) - 7/(18*(223/432 + sqrt(597)/48)^(1/3)) + 21/(4*sqrt(11/12 + 2*(223/432 + sqrt(597)/48)^(1/3) + 7/(18*(223/432 + sqrt(597)/48)^(1/3)))))/2))
Подстановка условия
[src]
x^4 - x^3 - x^2 - 2*x + 1 при x = -1/4
подставляем
4 3 2 x - x - x - 2*x + 1
$$x^{4} - x^{3} - x^{2} - 2 x + 1$$
4 2 3 1 + x - x - x - 2*x
$$x^{4} - x^{3} - x^{2} - 2 x + 1$$
переменные
x = -1/4
$$x = - \frac{1}{4}$$
4 2 3 1 + (-1/4) - (-1/4) - (-1/4) - 2*(-1/4)
$$(-1/4)^{4} - (-1/4)^{3} - (-1/4)^{2} - 2 (-1/4) + 1$$
373 --- 256
$$\frac{373}{256}$$
373/256