if-f-1-f-0-f-2-0-and-f-1-6-then-find-f-x-

Question Number 202795 by mathlove last updated on 03/Jan/24

i f f (- 1) = f (0) = f (2) = 0 a n d f (1) = 6

t h e n f i n d f (x) = ?

Answered by AST last updated on 03/Jan/24

I f f (x) i s a p o l y n o m i a l : x (x + 1) (x - 2) (x - 4)

Answered by mr W last updated on 03/Jan/24

f (- 1) = f (0) = f (2) = 0 :

\Rightarrow f (x) = x (x + 1) (x - 2) p (x)

f (1) = 6 :

\Rightarrow f (1) = 1 \times (1 + 1) (1 - 2) p (1) = 6

\Rightarrow p (1) = - 3

\Rightarrow p (x) = (x - 4) q (x) w i t h q (1) = 1

s o l u t i o n :

f (x) = x (x + 1) (x - 2) (x - 4) q (x)

w i t h q (x) = a n y f u n c t i o n a n d q (1) = 1 .

Commented by mathlove last updated on 03/Jan/24

t h a n k s m r W

Commented by Rasheed.Sindhi last updated on 03/Jan/24

f (x) = x (x + 1) (x - 2) \overset{?}{(x - 4)} q (x)

T h a n x i n a d v a n c e s i r!

Commented by mr W last updated on 03/Jan/24

s u c h t h a t f (1) = 6

Commented by Rasheed.Sindhi last updated on 03/Jan/24

T h a n X a g a i n s i r!

Answered by a.lgnaoui last updated on 03/Jan/24

- 1; 0 et 2 racines de f (x)

\Rightarrow f (x) = ax (x + 1) (x - 2)

f (1) = a \times 2 \times (- 1) = - 2 a = 6

\Rightarrow a = - \frac{6}{2} = - 3

alors f (x) = - 3 x (x + 1) (x - 2)

f (x) = - 3 x (x^{2} - x - 2)

Answered by Frix last updated on 04/Jan/24

Usually {we}^{'} re looking for the polynome

of minimal degree . With n equations we

get a polynome of degree n - 1 \Rightarrow with 4

equations we get a polynome of degree 3 .

The usual method is this :

{a x}^{3} + {b x}^{2} + c x + d = f (x)

f (0) = 0 \Rightarrow d = 0 (1)

f (- 1) = 0

- a + b - c = 0 (2)

f (2) = 0

8 a + 4 b + 2 c = 0 (3)

f (1) = 6

a + b + c = 6 (4)

(2) \Rightarrow b = a + c

(3) 12 a + 6 c = 0 \Rightarrow c = - 2 a \Rightarrow b = - a

(4) - 2 a = 6 \Rightarrow a = - 3 \Rightarrow b = 3 \land c = 6

\Rightarrow

f (x) = - 3 x^{3} + 3 x^{2} + 6 x = - 3 x (x - 2) (x + 1)

But with f (- 1) = f (0) = f (2) = 0 we directly

get

f (x) = a (x + 1) x (x - 2)

f (1) = 6 \Rightarrow a \times 2 \times 1 \times (- 1) = 6 \Rightarrow a = - 3

\Rightarrow

f (x) = - 3 x (x - 2) (x + 1)

Commented by mathlove last updated on 04/Jan/24

t h a n k s