Question

Use synthetic division to determine whether the number k is an upper or lower bound (as specified for the real zeros of the function f).

k = 2; f(x) = 2x3 + 4x2 + 2x - 4; Lower bound?

Answer 1

Ok just to clarify, I divide f(x) = 2x3 + 4x2 + 2x - 4 over k=2 using synthetic division?

meaning...the number in the corner will be -2?

Answer 2

2x3 + 4x2 + 2x - 4
0 4 16 36
--------------------
2 | 2 8 18 32



not sure about the k thing, -2 might be more useful indeed

Answer 3

what does "over k=2" mean?

Answer 4

I guess I meant /2...
to determine if its lower or upper bound i look at the signs right?

Answer 5

ive never applied synthD to bounds before. id have to read up on it to be sure

Answer 6

perhaps this is the setup ?\[f(x)=\frac{2x^3 + 4x^2 + 2x - 4}{x-k}\]

Answer 7

but how can I divide using synthetic division that way?

Answer 8

synthetic division is a method, and the setup is purely up to the user. I use a setup that i am comfortable with.

Answer 9

the remainder thrm tells us that the remainder of synthD gives us the value of f(k)

Answer 10

or at least the top portion of the rational function

Answer 11

when the remainder is zero, we know the value of k produces a factor of the top

Answer 12

The Upper and Lower Bound Thrms

Let f (x) be a polynomial with real coefficients and a positive leading
coefficient, and let a and b be nonzero real numbers.

1. Divide f (x) by x - b (where b > 0) using synthetic division. If the last row
containing the quotient and remainder has no negative numbers, then b is
an upper bound for the real roots of f (x) = 0.

2. Divide f (x) by x - a (where a < 0) using synthetic division. If the last row
containing the quotient and remainder has numbers that alternate in sign
(zero entries count as positive or negative), then a is a lower bound for
the real roots of f (x) = 0.

https://docs.google.com/viewer?a=v&q=cache:ms0PHZHMdkkJ:www.chesapeake.edu/khennayake/WebCT/blitzer/3.5.ppt+&hl=en&gl=us&pid=bl&srcid=ADGEEShoz8LO_eqsltMFlgGG_OijuHun1I30OqJr85ZDQH-CRaa4ntOeh0ER4SkQ9sN4p9WSaXMZ03XBwym7PoM58xf7wk3FSj9rc82aqX8bIxrHhh7wDtiRg_W5jMw_hvODDzlQDBl5&sig=AHIEtbSRef2pW6nl5rMTDpbvjThKNYVXbQ

Answer 13

so the last row has to be all positive values to be an upper bound; or all negative numbers to be a lower bound

that has something to do with the descartes signs

Answer 14

if a poly has all +s, then there is no positive root; if it has all negatives there is no negative root

Answer 15

well that would mean I have an upper bound then correct?

Answer 16

2x3 + 4x2 + 2x - 4
0 4 16 36
--------------------
2 | 2 8 18 32 <-- all positives, so k=2 is an upper bound. Might not be the most efficient upper bound, but yes

Answer 17

thanks for your help :) I think i got these now

Answer 18

lets try k=1

2x3 + 4x2 + 2x - 4
0 2 6 8
--------------------
1 | 2 6 8 4

it would appear the k=1 is an upper bound as well if we want to get closer and closer to it.

and youre welcome ;)

Answer 19

it has a root at about .69

http://www.wolframalpha.com/input/?i=+2x%5E3+%2B+4x%5E2+%2B+2x+-+4

so trying numbers above k=2 is pointless by the bounding thrms.