Question

Linear Algebra Question:

Find the polynomial of degree 4 whose graph goes through the points (-3, -227), (-1, -3), (0, 4), (1, 13), and (2, 18).

f(x) = ?

Answer 1

just plug in and you have your polynomial

Answer 2

thank you very much :)

Answer 3

\[f(x)=f(x_0) \cdot \frac{x-x_1}{x_0-x_1} \cdot \frac{x-x_2}{x_0-x_2} \cdot \frac{x-x_3}{x_0-x_3} \cdot \frac{x-x_4}{x_0-x_4}\]
\[+f(x_1) \cdot \frac{x-x_0}{x_1-x_0} \cdot \frac{x-x_2}{x_1-x_2} \cdot \frac{x-x_3}{x_1-x_3} \cdot \frac{x-x_4}{x_1-x_4}\]
\[+f(x_2) \cdot \frac{x-x_0}{x_2-x_0} \cdot \frac{x-x_1}{x_2-x_1} \cdot \frac{x-x_3}{x_2-x_3} \cdot \frac{x-x_4}{x_2-x_4}\]
\[+f(x_3) \cdot \frac{x-x_0}{x_3-x_0} \cdot \frac{x-x_1}{x_3-x_1} \cdot \frac{x-x_2}{x_3-x_2} \cdot \frac{x-x_4}{x_2-x_4}\]
\[+f(x_4) \cdot \frac{x-x_0}{x_4-x_0} \cdot \frac{x-x_1}{x_4-x_1} \cdot \frac{x-x_2}{x_4-x_2} \cdot \frac{x-x_3}{x_4-x_3}\]

Answer 4

Actually, I don't understand how to plug this in. the answer they are looking for is in the form of __(x^4) + __(x^3) + __(x^2) + __x + __

Answer 5

its too long to write on latex
are you sure you don't know how to plug in?

Answer 6

what is f(x_0)?

Answer 7

is it -227?

Answer 8

yes

Answer 9

what is x_1?

Answer 10

-

Answer 11

-3

Answer 12

x_1=-1

Answer 13

f(x_1)=-3

Answer 14

oh my bad. yes

Answer 15

f(x_1) is -3. read too fast

Answer 16

see you know how to use a formula lol

Answer 17

do you know what this is called?

Answer 18

its called lagrange polynomial

Answer 19

there may be other ways to find the polynomial but i remember lagrange best

Answer 20

no ive never seen anything so complexly put :P

Answer 21

lol

Answer 22

what way have you been using?

Answer 23

i think my teacher wanted it done with augmented matrices

Answer 24

i can't remember that way but if you want i can give you the answer using my way (well not my way, but by way lagrange)
just give a few minutes

Answer 25

ill do it too :)

Answer 26

http://www.wolframalpha.com/input/?i=227x%28x%2B1%29%28x-1%29%28x-2%29%2F%5B120%5D-3x%28x%2B3%29%28x-1%29%28x-2%29%2F%5B-12%5D%2B13%28x%2B3%29%28x%2B1%29x%28x-2%29%2F%5B-8%5D%2B18x%28x%2B3%29%28x%2B1%29%28x-1%29%2F%5B30%5D

Answer 27

i didn't feel like multiplying it all out and combining like terms lol

Answer 28

:D ty

Answer 29

Row reduce the matrix \[\left[\begin{matrix}(-3)^4 & (-3)^3 & (-3)^2 & -3 & 1 & -227
\\ (-1)^4 & (-1)^3 & (-1)^2 & -1 & 1 & -3
\\(0)^4 & (0)^3 & (0)^2 & 0 & 1 & 4
\\(1)^4 & (1)^3 & (1)^2 & 1 & 1 & 13
\\(2)^4 & (2)^3 & (2)^2 & 2 & 1 & 18\end{matrix}\right]\]

Answer 30

ty :)

Answer 31

do i consider (-3)^4 = -3 or 81?

Answer 32

you get

\[\left[\begin{matrix}1 & 0 & 0 & 0 & 0 & -2
\\ 0 & 1& 0& 0& 0& 3
\\0& 0 &1 & 0 & 0 & 3
\\0 & 0 & 0 & 1 & 0 & 5
\\0 & 0 & 0 & 0 & 1 & 4\end{matrix}\right]\]

Answer 33

\[(-3)^4=81\]

Answer 34

then \[f(x)=-2x^4+3x^3+3x^2+5x+4\]

Answer 35

wow that was fast :D. im still doing row operationS :P

Answer 36

tysvm

Answer 37

you get the same solution if you use the Lagrange Polynomial (as myininaya used)

Answer 38

i think i remember that way,zarkon lol

Answer 39

it is pretty simple...

let \[f(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0\]

plug in all five of the points given above into the function.

you get a system of 5 equations and 5 unknowns...then just solve the system.

Answer 40

when i let wolfram put into standard form I'm not getting what you have
-227x(x+1)(x-1)(x-2)/[120]-3x(x+3)(x-1)(x-2)/[-12]+4(x+3)(x+1)(x-2)(x-1)/[-6]+13(x+3)(x+1)x(x-2)/[-8]+18x(x+3)(x+1)(x-1)/[30]

Answer 41

http://www.wolframalpha.com/input/?i=-227x%28x%2B1%29%28x-1%29%28x-2%29%2F%5B120%5D-3x%28x%2B3%29%28x-1%29%28x-2%29%2F%5B-12%5D%2B4%28x%2B3%29%28x%2B1%29%28x-2%29%28x-1%29%2F%5B-6%5D%2B13%28x%2B3%29%28x%2B1%29x%28x-2%29%2F%5B-8%5D%2B18x%28x%2B3%29%28x%2B1%29%28x-1%29%2F%5B30%5D

Answer 42

you must have typed something in incorrectly. I have a Lagrange poly program on my calculator and it is giving the same result that i have above

Answer 43

so we have
(x,y)_0=(-3,-227)
(x,y)_1=(-1,-3)
(x,y)_2=(0,4)
(x,y)_3=(1,13)
(x,y)_4=(2,18)
?

Answer 44

those are the points i'm using is that what you have?

Answer 45

yes

Answer 46

dang it ...

Answer 47

http://www.wolframalpha.com/input/?i=-227x%28x%2B1%29%28x-1%29%28x-2%29%2F%28120%29-3x%28x%2B3%29%28x-1%29%28x-2%29%2F%28-12%29%2B4%28x%2B3%29%28x%2B1%29%28x-2%29%28x-1%29%2F%286%29%2B13%28x%2B3%29%28x%2B1%29x%28x-2%29%2F%5B-8%5D%2B18x%28x%2B3%29%28x%2B1%29%28x-1%29%2F%5B30%5D

Answer 48

i found your error...

Answer 49

i'm a horrible typer

Answer 50

me too

Answer 51

6 instead of -6

Answer 52

yep

Answer 53

ok but anyways both processes should yield the same answer (if you make no mistakes or typo's)
hopefully they will let you use a program for something like this
that's too much algebra by hand

Answer 54

when i was doing this stuff they made us make our own programs like in maple or something like that

Answer 55

but you have to understand what the process is to make the program

Answer 56

I used matlab..when I was taking numerical analysis

Answer 57

yep

Answer 58

i prefer maple

Answer 59

I don't like to program in maple

Answer 60

you like to program in your nspire

Answer 61

matlab is so easy to make programs with

Answer 62

yes I do

Answer 63

it has a nice programming language

Answer 64

i don't have enough money i will wait on the nspire :(

Answer 65

I got it as an x-mas gift

Answer 66

that is why christmas is great because of all the presents

Answer 67

my mother-in-law spends way too much money on me during Christmas :)

Answer 68

someone let me borrow their n-spire and i accidently dropped just once and i broke it so i have to buy them an n-spire lol
i will never borrow anything ever again from anyone

Answer 69

lol...that's not good

Answer 70

i had it for one freakin' day

Answer 71

crazy.

Answer 72

one drop=no work
wth

Answer 73

yeah...it's not like the 83...you drop the 83 and it will be fine

Answer 74

my ti83 i dropped that numerous times stills works
but still stolen

Answer 75

yep

Answer 76

so i'm getting a ti83 again
no upgrade

Answer 77

you should get an 83+...it has flash memory

Answer 78

actually i think it might have been plus

Answer 79

we will see what i get in the mail in a few days

Answer 80

i could go to my ebay account and look it up but i would rather be surprised :)

Answer 81

lol...nice

Answer 82

maybe they will put some really cool calculator program on the ipad one day

Answer 83

probably

Answer 84

that would be so cool to take your finger and point on the screen where you want the graph shaded or something like that

Answer 85

that would be nice

Answer 86

so algebra students won't have to think what is the inequality here: