Question

So lost with this if anyone can breakdown I would be so thankful :
Find polynomial of degree 4 whose graph goes through points (-3, -227) , (-1 , -3), (0 , 4), (1,13), and (2,18)

Answer 1

I know its a lot of work but i keep on messing up

Answer 2

\[(-3, -227) , (-1 , -3), (0 , 4), (1,13), (2,18)\] i do not think there is an easy way to do this, you will get 4 equations in 4 variables

Answer 3

Please , please just show me the breakdown and equation I have spent an hour on this

Answer 4

ok we have to solve for a, b , c and d here
\[ax^4+bx^2+cx+d\] replace x by 0 and we know \(d=4\) right off the bat

Answer 5

since \((0,4)\) is on the graph

Answer 6

so now we can reduce it to three variables and three equations
\[ax^4+bx^3+cx+4\] now replace x by -1 and set the result equal to 3

Answer 7

or rather -3

Answer 8

Yeah that part is easy ok I get that satellite you are great just keep going Ill let you know where I deviate

Answer 9

oh damn it is a 4th degree polynomial

Answer 10

Yeah.... and why set equal to -3?

Answer 11

lets start again
\[a_4x^4+a_3x^3+a_2x^2+a_1x+a_0\]

Answer 12

and if we put \(x=0\) we get \(a_0=4\) so we have \[a_4x^4+a_3x^3+a_2x^2+a_1x+4\]

Answer 13

Ok I see

Answer 14

now replace x by -1 and set the result equal to -3 because you are told that \((-1,-3)\) is on the graph
we get
\[a_4-a_3+a_2-a_2+4=-3\] or
\[a_4-a_3+a_2-a_1=7\] as one equation
4 more to go

Answer 15

jesuss.....

Answer 16

replace x by 1, set result equal to 13 get
\[a_4+a_3+a_2+a_1-4=13\] i.e.
\[a_4+a_3+a_2+a_1=17\]two more

Answer 17

you are the best satellite im gonna give u an essay of a best rating

Answer 18

let \(x=2\) and set equal to 18 get
\[16a_4+8a_3+4a_2+2a_1-4=18\] or
\[16a_4+8a_3+4a_2+2a_1=22\]

Answer 19

and i will review this problem i am so bad

Answer 20

(Why isn't it +4?)

Answer 21

ok maybe mertsj will come along with a snap way to do it

Answer 22

oh lord have mercy!

Answer 23

you have a clue how to do this merts?

Answer 24

I think just grind it out. I have found that a+c=1

Answer 25

And b + d = 8

Answer 26

\[a_4-a_3+a_2-a_2+4=-3\]
\[a_4-a_3+a_2-a_2=-7\]
\[a_4+a_3+a_2+a_1+4=13\]
\[a_4+a_3+a_2+a_1=9\]
\[16a_4+8a_3+4a_2+2a_1+4=18\]
\[16a_4+8a_3+4a_2+2a_1=14\]

Answer 27

one more to go
did you write the last one mertsj?

Answer 28

yeah i really need to review thanks for all this I know I am a bum...

Answer 29

ok finally replace \(x\) by -3 and get
\[81a_4-27a_3+9a_2-3a_1+4=-227\] so
\[81a_4-27a_3+9a_2-3a_1=-231\]

Answer 30

so finally we have our 4 equations and we need to solve the system of equations

Answer 31

\[a_4-a_3+a_2-a_1=-7\]\[a_4+a_3+a_2+a_1=9\]\[16a_4+8a_3+4a_2+2a_1=14\]\[81a_4-27a_3+9a_2-3a_1=-231\]

Answer 32

feel like solving it?

Answer 33

Whattt i thought this was final equation dammnnn

Answer 34

So we are solving that final equation 81a4... = -231

Answer 35

?

Answer 36

no we have to solve the entire system of equations
four variables, for equations

Answer 37

oh.

Answer 38

can u show me how to do it? i am like totally lost if you start and I get it I'll stop u

Answer 39

and here is the solution!!
http://www.wolframalpha.com/input/?i=a4%E2%88%92a3%2Ba2%E2%88%92a1%3D%E2%88%927%2C+a4%2Ba3%2Ba2%2Ba1%3D9+%2C16a4%2B8a3%2B4a2%2B2a1%3D14%2C+81a4%E2%88%9227a3%2B9a2%E2%88%923a1%3D%E2%88%92231

Answer 40

so how would we write the equation?

Answer 41

it is a 4 by 4 system of equations which is why they invented technology.
doing it by hand is for donkeys

Answer 42

like what would the coefficient of x^4 be?

Answer 43

\[a_0=4,a_1 = 5, a_2 = 3, a_3 = 3, a_4 = -2\] so we have it

Answer 44

\[-2x^4+3x^3+3x^2+5x+4\] is our solution

Answer 45

So it is once again made clear why you are the sole guru!! Well done!

Answer 46

maybe mertsj has a quick way to do it by hand, but i really doubt it

Answer 47

Nope.

Answer 48

i will take the compliment (thank you) but i just copied and pasted from wolfram
helps if you use what you got to get what you want

Answer 49

Love you satellite :)

Answer 50

merci

Answer 51

:)