Практикум по алгебре. Часть 1. Многочлены и их корни. Попов В.В - 10 стр.

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f(x) x c
a) f(x) = 3x
6
4x
5
4x
4
+ 2x
3
3x
2
+ 2x 3, c = 2;
b) f(x) = x
6
+ 2x
5
3x
4
+ 5x
3
5x
2
3x 4, c = 1;
c) f(x) = x
6
+ 3x
5
3x
4
5x
3
+ 5x
2
1, c = 3;
d) f(x) = x
6
5x
5
+ 2x
4
+ 4x
3
+ 5, c = 3;
e) f(x) = x
6
+ 5x
5
+ 4x
4
+ 2x
3
+ 4x
2
+ 2x 2, c = 3.
f(x) c
a) f(x) = 2x
6
+ 4x
5
x
3
4x 4, c = 2;
b) f(x) = x
6
3x
5
+ 3x
3
+ 2x
2
4x + 5, c = 2;
c) f(x) = x
6
5x
5
+ 4x
4
+ 4x
3
+ 5x
2
3x + 2, c = 2;
d) f (x) = x
6
5x
5
+ 5x
4
3x
3
4x
2
2x + 2, c = 2;
e) f(x) = x
6
+ 4x
5
+ 2x
4
+ 3x
2
2x, c = 3;
f) f(x) = x
6
+ 3x
5
2x
4
+ 4x
3
+ 5x
2
x 5, c = 3.
f(x)
xc
c
a) f(x) = x
6
+ 2x
5
+ x
4
+ 5x
3
+ 3x
2
+ 2x + 5, c = 2;
b) f(x) = x
6
+ 5x
5
+ 4x
4
5x
3
5x
2
+ 5x + 4, c = 3;
c) f(x) = 3x
6
x
5
5x
4
+ x
3
+ 2x
2
3x + 3, c = 1;
d) f (x) = x
6
4x
5
+ 3x
3
+ 3x
2
+ 1, c = 3;
e) f(x) = x
6
5x
5
+ 5x
4
+ 4x
3
4x
2
5, c = 3;
f) f(x) = x
6
4x
5
+ x
4
+ x
2
4x, c = 3.
1)
x
3
+ 2x 1
(x 2)
4
; 2)
x
4
2x
3
+ 1
(x + 1)
5
; 3)
x
3
+ 3x + 2
(x + 1)
4
.
   Çàäà÷è.

   2.1. Èñïîëüçóÿ ñõåìó Ãîðíåðà, ðàçäåëèòü ñ îñòàòêîì ìíî-
ãî÷ëåí f (x) íà x − c:

   a)    f (x) = 3x6 − 4x5 − 4x4 + 2x3 − 3x2 + 2x − 3,      c = 2;
   b)    f (x) = x6 + 2x5 − 3x4 + 5x3 − 5x2 − 3x − 4,       c = −1;
   c)    f (x) = x6 + 3x5 − 3x4 − 5x3 + 5x2 − 1,            c = −3;
   d)    f (x) = x6 − 5x5 + 2x4 + 4x3 + 5,                  c = 3;
   e)    f (x) = x6 + 5x5 + 4x4 + 2x3 + 4x2 + 2x − 2,       c = −3.
    2.2. Èñïîëüçóÿ ñõåìó Ãîðíåðà, íàéòè çíà÷åíèå ìíîãî÷ëåíà
f (x) â òî÷êå c:

   a)        f (x) = 2x6 + 4x5 − x3 − 4x − 4,               c = −2;
   b)        f (x) = x6 − 3x5 + 3x3 + 2x2 − 4x + 5,         c = 2;
   c)        f (x) = x6 − 5x5 + 4x4 + 4x3 + 5x2 − 3x + 2,   c = 2;
   d)        f (x) = x6 − 5x5 + 5x4 − 3x3 − 4x2 − 2x + 2,   c = 2;
   e)        f (x) = x6 + 4x5 + 2x4 + 3x2 − 2x,             c = −3;
   f)        f (x) = x6 + 3x5 − 2x4 + 4x3 + 5x2 − x − 5,    c = −3.
   2.3. Èñïîëüçóÿ ñõåìó Ãîðíåðà, ìíîãî÷ëåí f (x) ðàçëîæèòü
ïî ñòåïåíÿì x − c. Íàéòè çíà÷åíèÿ âñåõ ïðîèçâîäíûõ ýòîãî ìíî-
ãî÷ëåíà â òî÷êå c:
   a)        f (x) = x6 + 2x5 + x4 + 5x3 + 3x2 + 2x + 5,    c = −2;
   b)        f (x) = x6 + 5x5 + 4x4 − 5x3 − 5x2 + 5x + 4,   c = −3;
   c)        f (x) = 3x6 − x5 − 5x4 + x3 + 2x2 − 3x + 3,    c = −1;
   d)        f (x) = x6 − 4x5 + 3x3 + 3x2 + 1,              c = 3;
   e)        f (x) = x6 − 5x5 + 5x4 + 4x3 − 4x2 − 5,        c = 3;
   f)        f (x) = x6 − 4x5 + x4 + x2 − 4x,               c = 3.

   2.4. Ðàçëîæèòü ñëåäóþùèå äðîáè íà ïðîñòåéøèå:
              x3 + 2x − 1      x4 − 2x3 + 1      x3 + 3x + 2
        1)                ; 2)              ; 3)             .
                (x − 2)4         (x + 1)5          (x + 1)4

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