Question

What are the zeros of the function x^4 -5 x^3 + 13x - 7?

Answer 1

try the easy numbers first
easiest is 1, but \(1-5+13-7=2\) so that is not right
then try \(-1\) and get \(1+5-13-7\) bingo

Answer 2

then to find the others, start by factoring \((x+1)\) out of your equation

Answer 3

so is then one of the zeros is -1?

Answer 4

yes

Answer 5

oh no!!
i made a mistake

Answer 6

oh no! lol what is it?

Answer 7

\(-1\) is not a zero

Answer 8

make sure you posted the correct problem
the zeros are not rational, no good way to find them

Answer 9

are you sure it ends with \(-7\) and not \(+7\) ?

Answer 10

well the question was create your own polynomial with a degree greater than 2. Attach the graph to the word document and find the zeros of the function. I dont really know how to do it lol

Answer 11

well you picked a rather lousy one
you get to make up your own?

Answer 12

i can show you how to pick a nice one if you like

Answer 13

yes please!

Answer 14

pick the zeros and work backwards
for example, if you want the zeros to be \(-2, 1, 3\) then start with
\[(x+2)(x-1)(x-3)\] and multiply out
that will give you the zeros you want, and be easy to graph

Answer 15

so i just multiply those and it will give me a function?

Answer 16

yes
that will give you a polynomial of degree 3 with those zeros
even easier to cheat by doing this
http://www.wolframalpha.com/input/?i=%28x%2B2%29%28x-1%29%28x-3%29

Answer 17

that multiplies out for you
you see you get \(x^3-2 x^2-5 x+6\)
it also gives you a nice graph to copy (the second one)

Answer 18

omg thank you so much!

Answer 19

do you think you could check my work on the rest of the assignment. I think i did it right but i just want to be sure

Answer 20

go ahead and post it i will look

Answer 21

thank you! just a sec!

Answer 22

Graph the function f(x) = (x + 3)3 by hand and describe the end behavior. (1 point)
Graph the function f(x) = –x4 – 4 by hand and describe the end behavior. (1 point)
Graph the function f(x) = –3x3 + 9x2 – 2x + 3 using graphing technology and describe the end behavior. (1 point)
Graph the function f(x) = x4 – 7x3 + 12x2 + 4x – 12 using graphing technology and describe the end behavior. (1 point)
Without using technology, describe the end behavior of f(x) = –3x38 + 7x3 – 12x + 13. (1 point)
Using complete sentences, explain how to find the zeros of the function f(x) = 2x3 – 9x + 3. (2 points)
Create your own polynomial with a degree greater than 2. Attach the graph to the word document and find the zeros of the function. (3 points)

Answer 23

3. Graph the function f(x) = –3x3 + 9x2 – 2x + 3 using graphing technology and describe the end behavior. (1 point)
• This is a cubic function. You can determine that the ends go in different directions because the degree is an odd number. The line crosses the x-axis at -1/2 and crosses the y-axis at 4. It has a positive slope.


4. Graph the function f(x) = x4 – 7x3 + 12x2 + 4x – 12 using graphing technology and describe the end behavior. (1 point)
• This is a quartic function. The line crosses the x-axis at -9 and then crosses y-axis at 1. It has a positive slope.

5. This is a cubic function. It has a very steep negative slope. It crosses the y axis at 13 and the x as at 0.
6. You could graph the function and find the x intercepts or factor the equation and solve for f(x) = 0

ill post the graphs in a second

Answer 24

you can check your answers by looking at wolfram
first one looks just like \(y=x^3\) but shifted left 3 units

Answer 25

okay, thanks!

Answer 26

it even tells you to use graphing technology, so that is not cheating

Answer 27

for example, the third one looks like this
http://www.wolframalpha.com/input/?i=+%E2%80%933x3+%2B+9x2+%E2%80%93+2x+%2B+3

Answer 28

end behaviour, starts at \(\infty\) and then goes to \(-\infty\)

Answer 29

which graph is it?

Answer 30

look at the top, you will see which one i graphed
i think it is #3

Answer 31

there is a mistake in your answer here
4. Graph the function f(x) = x4 – 7x3 + 12x2 + 4x – 12 using graphing technology and describe the end behavior. (1 point) • This is a quartic function. The line crosses the x-axis at -9 and then crosses y-axis at 1. It has a positive slope.

Answer 32

it is not a line, and and therefore does not have a positive slope

Answer 33

end behaviour, as \(x\to \infty\) you have \(f(x)\to \infty\) and also as \(x\to -\infty\) you have \(f(x)\to \infty\)

Answer 34

similar mistake here
5. This is a cubic function. It has a very steep negative slope. It crosses the y axis at 13 and the x as at 0.

Answer 35

it is a cubic function, but since it is not a line it does not have a slope

Answer 36

actually, i am not sure which function #5 was referring to

Answer 37

number 5 is f(x) = –3x38 + 7x3 – 12x + 13.

Answer 38

it it really \(-x^{38}+7x^3-12x+13\) ???

Answer 39

because that is a polynomial of degree 38, not degree 3

Answer 40

yeah!

Answer 41

ok then change your answer
the degree is 38 so it is not cubic

Answer 42

so its quartic?

Answer 43

because the degree is even, and the leading coefficient is negative, you know as \(x\to -\infty\) you have \(f(x)\to -\infty\) and as \(x\to \infty\) also \(f(x)\to -\infty\)

Answer 44

oh no

Answer 45

quartic means degree is 4

Answer 46

quintic is 5

Answer 47

so if its to the 38th degree what is it?

Answer 48

after that you just says "nth degree" for for this one it is a polynomial of degree 38

Answer 49

no special term for it

Answer 50

okay so then whats the end behavior? Sorry, Im just really bad at math lol

Answer 51

lol no problem
end behaviour is what i wrote above, for the reason i wrote also

Answer 52

so what kind of function is it again?

Answer 53

lets cut to the chase
if the degree is even, and the leading coefficient is positive, like \(x^4\) then the end behaviour is this: as \(x\to -\infty\) and as \(x\to \infty\) you have \(f(x)\to \infty\)

Answer 54

ohh okay

Answer 55

if the degree is even, and the leading coefficient is negative, like say \(-x^6\) then the end behaviour is this: as \(x\to \infty\) and as \(x\to -\infty\) you have \(f(x)\to -\infty\)

Answer 56

that is the case for the one that started with \(-x^{38}\)

Answer 57

so the end behavior for that is x→∞ and as x→−∞ you have f(x)→−∞

Answer 58

now if the degree is odd, and the leading coefficient is positive, like say \(x^3\) then the end behaviour is this: as \(x\to -\infty\) you have \(f(x)\to -\infty\) and as \(x\to \infty\) you ahve \(f(x)\to \infty\)

Answer 59

because its negative and even

Answer 60

right

Answer 61

okayyy, that makes a lot more sense

Answer 62

and finally, the last case, if the degree is odd and the leading coefficient is negative, like say \(-x^7\) then as \(x\to -\infty\) you have \(f(x)\to \infty\) and as \(x\to \infty\) you have \(f(x)\to -\infty\)

those are the 4 cases