f

f(x) = a0 + a1x+ a2x2 + · · ·+ anxn,

ai ai fn

f ain

aai

ai,j

ff

fx a n

a0, a1, a2, . . . , an n

g tb n = 3 b0, b1, b2, b3 = 2, 0, 4,−1

g

g(t) = 2 + 0t+ 4t2 + (−1)t3.

g g(t) = 2+4t2− t3g t = 2

2 t

g(2) = 2 + 4(22)− 23 = 10.

g(t) = (t− 1)(t+ 6)2

t1, t2, t3, . . . , tn

f(x) = x+ 1g(t) = 1 + t

t f(t)(t, f(t))

f(x) = x5 − x− 1

3 6

(x, y) = xy

(x) = 1− x(x, y) = 1− (1− x)(1− y)

f(x) = 0

g(x) = 12

h(x) = 1 + x+ x2 + x3

i(x) = x1/2

j(x) =1

2+ x2 − 2x4 + 8x8

k(x) = 4.5 +1

x− 5

x2

l(x) = π − 1ex5 + eπ3x10

m(x) = x+ x2 − xπ + xe

n! = 1 ·2 · · · · ·n 0! = 1

f(x) = 0f

n = 0 a0 = 0f n = 1 a0 = 0 a1 = 0 n = 2

f(x) = 0

anf(x) = 0

f(x) = 0−1

f : A → B

f : A → BA B f

f : A→ B

R

f : R → R

ZN

∈q ∈ N qq q

p, p+ 2 11 13 23

n ≥ 0 n + 1(x1, y1), (x2, y2), . . . , (xn+1, yn+1) R2 x1 < x2 < · · · < xn+1

p(x) n p(xi) = yi i

R2Z3 N10

nn + 1

n = 0 n+ 1 = 1(7, 4)

a0+a1x+a2x2+ · · ·+adxdd n d = 0d = 0 f f(x) = a0

f(7) = 4 f(x) = 4

(7, 4)

n = 1 n + 1 = 2(2, 3), (7, 4) f f(2) = 3f(7) = 4

f(x) = a0 + a1x.

f(2) = 3, f(7) = 4

a0 + a1 · 2 = 3a0 + a1 · 7 = 4

f(x) (a, b) f(a) = bf

a0 a0 = 3− 2a1(3 − 2a1) + a1 · 7 = 4 a1 = 1/5

a0 = 3− 2/5

f(x) =

(3− 2

5

)+

1

5x =

13

5+

1

5x.

x1 < x2 < · · · < xn+1 x1 < x2x1 = x2

(2, 3), (2, 5)

a0 + a1x

f(2) = 3 f(2) = 5

a0 + a1 · 2 = 3a0 + a1 · 2 = 5

n = 2

(x1, y1) n = 0 x1y1

f(x1) = y1 ff y1

f(x) = y1

(x1, y1), (x2, y2) xx1, x2, . . .

f(x) = y1x− x2x1 − x2

+ y2x− x1x2 − x1

f x1 x1 − x1 =0y1

x1−x2x1−x2 = y1 · 1 x1 y1

x1 = x20/0f(x2)

y2 f(x1) = y1 f(x2) = y2f

f

f(x) =y1

x1 − x2(x− x2) +

y2x2 − x1

(x− x1),

f(x) =x1y2 − x2y1x1 − x2

+

(y1 − y2x1 − x2

)x

xf x

f(x1) = y1(x1, y1), (x2, y2), (x3, y3)

x1

x2

f(x) = y1(x−x2)(x−x3)

(x1−x2)(x1−x3) + y2(x−x1)(x−x3)

(x2−x1)(x2−x3) + y3(x−x1)(x−x2)

(x3−x1)(x3−x2)

f x1y1

x2, x3f 2

n (x1, y1), . . . , (xn, yn)i yi

x−xj j i(xi − xj) j

f(x) =n∑

i=1

yi ·

⎛

⎝∏

j ̸=i

x− xjxi − xj

⎞

⎠

∑,∏

∑ni=1( )

∑ni=0

n+ 1

∑

+ ∑ ∏

∏j ̸=i i

jj

ji i

j xx 1 n j

j ̸= i jj = i

∏j ̸=i

j i

f(x) =n∑

i=1

(i)

⎛

⎝∏

j ̸=i(i, j)

⎞

⎠

∑ ∏

(x1, y1), . . . , (xn+1, yn+1) n + 1 xif(x)

f(x) =n+1∑

i=1

yi∏

j ̸=i

x− xjxi − xj

f(x) nn i xi ii yi

!

f : R→ R z f(z) = 0

R nn

f g n(x1, y1), . . . , (xn+1, yn+1)

f gf, g (f −g)(x)(f − g)(x) = f(x) − g(x) f − g

f ai g bi f − gci = ai − bi f g ci ai −bi

f, g f − gf − g n

x7

x5 (f − g)(xi) = 0 i xxi

(x1, y1), . . . , (xn+1, yn+1) f g if(xi) = g(xi) = yi

d f − g d ≤ nf−g n

n+1 xi f − g f − gf g

n ≥ 0 n + 1(x1, y1), (x2, y2), . . . , (xn+1, yn+1) R2 x1 < x2 < · · · < xn+1

p(x) n p(xi) = yi i

(x1, y1), . . . , (xn+1, yn+1) xif(x)

f(x) =n+1∑

i=1

yi

⎛

⎝∏

j ̸=i

x− xjxi − xj

⎞

⎠

f(x) ≤ nn i xi ii yi

g(x)f = g f−g

n n+ 1 xif − g f = g

nn+ 1

x

(x − xj)/(xi − xj)a0 = −xj/(xi − xj) a1 = 1/(xi − xj)

nk ≤ n

s f(x) f(0) =s d f(x) d f

a0, . . . , ad f a0 = sad ̸= 0 n f(x)

f(1), f(2), . . . , f(n) i (i, f(i))k

k − 1 g(x)g(x) f(x)

g(0) = f(0) dk

d = k − 1 kg(x) g(x) = f(x)

n = 5k = 3 f(x)

d = k − 1 = 2 109 f

f(x) = 109 + · x+ · x2

(1, f(1)), (2, f(2)), (3, f(3)), (4, f(4)), (5, f(5))

f(x) = 109− 55x+ 271x2,

f(0) = 109

(1, 325), (2, 1083), (3, 2383), (4, 4225), (5, 6609).

f(x)

f(x) f(0) kf(0)

f f(0) k

f d d

d dx x = 0 y (x, y)

y

df(0) y

y f(0)s =

f(x) f(0)s f(1), . . . , f(10) y

10f(1), f(2), . . . , f(10)y = f(0) 10

109

f(x) = 109− 55x+ 271x2

(2, 1083), (5, 6609)533 f(2) = 1083 f(5) =6609 f(0) = 533

(2, 1083), (5, 6609)

f 2 g 1f · g 3

f n g mf · g n+m

f g−1

a, bn ϕ(n) n > 1

n nn > 1

an 1

f g f(x) =g(x)h(x) h f g h

g hf

f, gf g

a, naϕ(n) n

x

√2 π e

φ = 1+√5

2

√2 +√3

π

e π + e πe

f(x) = a0 + a1x+ · · ·+ anxn n nr1, . . . , rn

n∑

i=1

ri = −an−1an

n∏

i=1

ri = (−1)na0an

.

r f(x) f(x) = (x−r)g(x)g(x)

x y

ff

w(x) =20∏

i=1

(x− i)

x19 w(x) −210 2−230.5

p

M > 1M = m1 ·m2 · · ·mk mi > 1

i, j mi mj r1, . . . , rk0 ≤ ri < mi ri mi

x 0 ≤ x < M x = ri mi i

kp(x) = (x − a1)(x − a2) · · · (x − ak)

ai

2

p(x) = (x− a1)(x− a2) · · · (x− am)(x2 + bm+1x+ am+1) · · · (x2 + bkx+ ak),

f(x) g(x) h(x) j(x) l(x) i√x = x1/2 k(x)

m(x)π, e π e

p, p + 2

M pq p q − p ≤M

M 26

100 M 70

M246

6

f

S

d d(x, y)x, y S

S S

dd(x, y) =

d(y, x)

f

fd

S f d

f (S, d) d

n p np p

a = n p an/p n < p n p = n

≡

a ≡ n p.

pp

n p k (n · k) ≡ 1 p

kf(x) x p

0 p

f(x) x pf

xf(0)

(d+ 1)f(0) 0 p