Практикум по алгебре. Часть 1. Многочлены и их корни. Попов В.В - 8 стр.

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f(x) n
x c
f(x) = b
0
(x c)
n
+ b
1
(x c)
n1
+ . . . + b
n1
(x c) + b
n
,
c
b
n
f(x) xc b
n1
x c
f(x) g(x)
1) f(x) = 4x
5
+ 10x
4
+ 8x
3
+ 5x
2
+ 2x + 3,
g(x) = 2x
3
+ 3x
2
x;
2) f(x) = 3x
5
+ 4x
4
6x
3
2x
2
+ 4x 2,
g(x) = 3x
3
2x
2
2x;
3) f(x) = x
5
+ 4x
4
2x
3
7x
2
+ 3x + 5,
g(x) = x
3
2x;
4) f(x) = 3x
5
5x
4
6x
3
+ 6x
2
+ 6x + 5,
g(x) = 3x
3
2x
2
2x;
5) f(x) = 6x
5
4x
4
+ 2x
3
8x
2
+ 6x 2,
g(x) = 3x
3
+ x
2
+ 2x 2;
6) f(x) = 6x
5
+ x
4
+ 3x
3
+ 3x
2
x + 4,
g(x) = 3x
4
x
3
+ 2x
2
;
7) f(x) = x
6
+ 4x
5
+ 3x
4
9x
3
5x
2
+ 7x + 1,
g(x) = x
3
+ 4x
2
+ 5x 2;
8) f(x) = 3x
5
+ x
4
+ 4x
3
+ 3x 1,
g(x) = 3x
4
+ x
3
+ 4x
2
x + 4.
1) q(x) = 2x
2
+ 2x + 2, r(x) = x
2
+ 4x + 3;
2) q(x) = x
2
+ 2x, r(x) = 2x
2
+ 4x 2;
3) q(x) = x
2
+ 4x, r(x) = x
2
+ 3x + 5;
4) q(x) = x
2
x 2, r(x) = 2x + 5;
5) q(x) = 2x
2
2x, r(x) = 2x 2;
6) q(x) = 2x + 1, r(x) = x
2
x + 4;
7) q(x) = x
3
2x + 1, r(x) = x
2
2x + 3;
8) q(x) = x, r(x) = x
2
x 1.
   1.9. Èçâåñòíî, ÷òî ëþáîé ìíîãî÷ëåí f (x) ñòåïåíè n ìîæíî
ðàçëîæèòü ïî ñòåïåíÿì ðàçíîñòè x − c:
    f (x) = b0 (x − c)n + b1 (x − c)n−1 + . . . + bn−1 (x − c) + bn ,
ãäå c ëþáîå ÷èñëî  âåùåñòâåííîå èëè êîìïëåêñíîå. Ïîêàçàòü,
÷òî bn ÿâëÿåòñÿ îñòàòêîì îò äåëåíèÿ f (x) íà x−c, bn−1  îñòàòîê
îò äåëåíèÿ íåïîëíîãî ÷àñòíîãî íà x − c, è ò.ä.
   1.10. Ðàçäåëèòü ñ îñòàòêîì ìíîãî÷ëåí f (x) íà g(x) â ñëå-
äóþùèõ ñëó÷àÿõ:

1) f (x) = 4x5 + 10x4 + 8x3 + 5x2 + 2x + 3,
   g(x) = 2x3 + 3x2 − x;
2) f (x) = 3x5 + 4x4 − 6x3 − 2x2 + 4x − 2,
   g(x) = 3x3 − 2x2 − 2x;
3) f (x) = x5 + 4x4 − 2x3 − 7x2 + 3x + 5,
   g(x) = x3 − 2x;
4) f (x) = 3x5 − 5x4 − 6x3 + 6x2 + 6x + 5,
   g(x) = 3x3 − 2x2 − 2x;
5) f (x) = 6x5 − 4x4 + 2x3 − 8x2 + 6x − 2,
   g(x) = 3x3 + x2 + 2x − 2;
6) f (x) = 6x5 + x4 + 3x3 + 3x2 − x + 4,
   g(x) = 3x4 − x3 + 2x2 ;
7) f (x) = x6 + 4x5 + 3x4 − 9x3 − 5x2 + 7x + 1,
   g(x) = x3 + 4x2 + 5x − 2;
8) f (x) = 3x5 + x4 + 4x3 + 3x − 1,
   g(x) = 3x4 + x3 + 4x2 − x + 4.
    Îòâåòû:     1)   q(x) = 2x2 + 2x + 2,     r(x) = x2 + 4x + 3;
                2)   q(x) = x2 + 2x,          r(x) = 2x2 + 4x − 2;
                3)   q(x) = x2 + 4x,          r(x) = x2 + 3x + 5;
                4)   q(x) = x2 − x − 2,       r(x) = 2x + 5;
                5)   q(x) = 2x2 − 2x,         r(x) = 2x − 2;
                6)   q(x) = 2x + 1,           r(x) = x2 − x + 4;
                7)   q(x) = x3 − 2x + 1,      r(x) = x2 − 2x + 3;
                8)   q(x) = x,                r(x) = x2 − x − 1.

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