Question

construct a polynomial of degree 4 that touches the x axis at -3 and stays on the positive side of the axis and also goes through the x axis at 2 and 5. Also determine the constant multiplier that makes the function have (1,192) on the graph.

Answer 1

what will be your 4 roots, since a degree 4 needs 4 roots

Answer 2

how do I figure that out?

Answer 3

you need to see where it ineracts with the x axis

Answer 4

that would be -3, 2 and 5 ?

Answer 5

but Im not sure about the 4th

Answer 6

yes, but with a little modification

think of "just touching" as going into the axis at -3, and out of the axis at -3, giving us a multiple root at -3

roots: -3,-3,2,5

Answer 7

oh ok. that makes sense

Answer 8

now make a product with your roots, by subtracting "x" from each root. tell me if that makes any sense

Answer 9

-3x^4-3x^3+2x^2+5x

Answer 10

well, that might be correct, but its a step or 2 away

Answer 11

does the product of the roots make sense?

Answer 12

(-3-x) (-3-x) (2-x) (5-x)

Answer 13

I guess Im not sure what you mean

Answer 14

do the products need to equal something thats the same?

Answer 15

the "multiply by zero" property makes y=0 whenever we use a root.

2
(-3-x) (-3-x) (2-x) (5-x) = 0
0


-3
(-3-x) (-3-x) (2-x) (5-x) = 0
0


-3
(-3-x) (-3-x) (2-x) (5-x) = 0
0

5
(-3-x) (-3-x) (2-x) (5-x) = 0
0

by subtracting x from each root, and multiplying the terms together, we get a polymonial

Answer 16

y = (-3-x) (-3-x) (2-x) (5-x)

y = (x^2-6x+9) (x^2-7x +10)

y = x^4 - x^3 - 23x^2 - 3x + 90 ; this will be your basic polynomial with the given roots

but we have to make one more modification ...

Answer 17

we do not want to "add" anything to the basic structure, otherwise we alter the root: 0+b = b. and not 0

so we can only multiply this set up to modify it to meet our needs

Answer 18

ok

Answer 19

what do we get when x = 1 in our basic setup?

y = 1 - 1 - 23 - 3 + 90 = 64

since we want x=1, y=192, and we can only multiply to modify this we have

64n = 192

n = 192/64 = 3

so lets multiply the basic setup by 3:

y = 3x^4 - 3x^3 - 69x^2 - 9x + 270 is the poly we desire

Answer 20

its actually 1192

Answer 21

lol, same process, different "n"

Answer 22

the point in question is: (1, 192) ? or, (1, 1192)?

Answer 23

you're right 1, 192

Answer 24

yay!! i can stave of senility for another day :)

Answer 25

lol

Answer 26

a textbook method is similar to mine, except they propose you subtract the root from x to make the product

(x--3)(x--3)(x-2)(x-5)

(x+3)(x+3)(x-2)(x-5) ... gets you the same answer, but i find the subtract a negative and switch to a plus a little useless .... just an added step is all

Answer 27

ok cool thank you

Answer 28

good luck ;)

Answer 29

do you know how I would determine the coordinates of the turning points?

Answer 30

yes

Answer 31

you know one of the turning points already (-3,0) has to be a turning point since it only touches the line and doesnt go thru it.

Answer 32

are you familiar with derivatives? from calculus?