Question

For the function f(x) = 3x^84 – 4x^43 + x7 – 4

a. What is the remainder when the function is divided by x – 1.

b. Is (x + 1) a factor of the function? Why or why not?

c. Is there a zero to the function between -1 and 1? Why or why not?

I'm not sure how I would do this? Something along the lines of synthetic division?

Answer 1

a. -4 (x-1) is a factor of f(x) without the free term
b. no, f(-1) != 0
c. f(-1)>0 , f(1)<0 , from the intermediate value theorem, there exists such c between -1, and 1 such that f(c)=0

Answer 2

Okay so how did you figure that out?
a. For example I don't know how to do that.
b. Do you just plug (x+1) in to X? How would I do that?
c. from the intermediate value theorem, there exists such c between -1, and 1 such that f(c)=0
could you go into more detail about that

Answer 3

a. if you just plug x=1 to 3x^84-4x^43+x^7 you get 0, so 1 is a solution for the equation and thus (x-1) is a divisor for it, so of course -4 (the free term) is the remainder.
(I can right down every equation for instance x^2-3x+2 = (x-2)(x-1) )
b. no, I plugged x=-1, and didn't get 0 - so (x+1) isn't a divisor
c. if f(-1)>0 and f(1)<0 then there has to be a value between -1 and 1 that gives 0 , otherwise the function "jumps" over 0, but this function is continuous (becuase it's a polynomial) so it can't "jump" . read here : http://en.wikipedia.org/wiki/Intermediate_value_theorem

Answer 4

hmm wasn't to coherent there - I meant I can write down every equation with it's divisors and it's remainder , for instance ; x^2-3x+2 = (x-2)(x-1)