Question

Part 1: Find the polynomial f(x) that has the roots of –3, 5 of multiplicity 2. (4 points)

Part 2: Explain how you would verify the zeros of f(x). (4 points)

Answer 1

If you have a collection of roots \(r_1, r_2,...,r_n\) and want to construct the polynomial, write a series of product terms \[P(x) = (x-r_1)(x-r_2)...(x-r_n)\]If a root has multiplicity > 1, you need to repeat it that many times in the product. You'll probably be expected to expand that product.

To verify the zeros, one way would be to check that the polynomial will have a value of 0 at each zero. Another would be to divide the polynomial by \((x-r_n)\) (to verify that \(r_n\) is a zero) — the division will have a remainder of 0 is \(r_n\) is a zero of the polynomial.

Answer 2

Another way, of course, would be to graph the polynomial...

Answer 3

The value of \(x\) where the curve touches or crosses the x-axis will be zeros of the polynomial.

Answer 4

so its (x+3) (x-5) (x-2) ? @whpalmer4

Answer 5

Nope, reread what I said about multiplicity.

Answer 6

dont you foil now ? @whpalmer4

Answer 7

Also, the problem is poorly worded: there are infinitely many polynomials with those zeros, because you can have a constant multiplier in front of all of the product terms without changing the zeros. It is inappropriate to speak of "the polynomial with zeros" unless there is a specified point which the graph of the polynomial goes through, which then uniquely identifies that constant multiplier. The problem should have said "find A polynomial ..."

Answer 8

Could you make me a list of all of the roots?

Answer 9

i dont understand

Answer 10

you need a list of all of the zeros (aka roots) to do this. what are they?

Answer 11

" polynomial f(x) that has the roots of –3, 5 of multiplicity 2" what does that mean to you?

Answer 12

it means they made it equal to zero. so it used to be (x+3) (x-5) (x-2)

Answer 13

nope. make me a list of roots. x = ?

Answer 14

"roots of -3, 5 of multiplicity 2" - what does that mean?

Answer 15

x=-3
x=5
x=2

Answer 16

okay, what does multiplicity mean?

Answer 17

i dont know

Answer 18

hint: "multiplicity 2" does not necessarily mean that x=2 is a root/zero!

Answer 19

multiply ?

Answer 20

okay, here's a tip: if you don't understand all the words in the problem, you probably aren't going to get the right answer!

Answer 21

multiplicity means the number of times a root/zero is repeated. for example,
\[y=x^2+2x + 1\] has a root of \(x=1\) with multiplicity 2 because it can be factored as \(x^2+2x+1 = (x+1)(x+1)\)

Answer 22

If a root has an odd multiplicity, the curve crosses the x-axis at the zero/root. If the root has an even multiplicity, the curve either dips down to or climbs up to and touches the x-axis at the root, then retreats. For example, \(y = x^2\) has multiplicity 2, and as it is a parabola, the curve drops to the x-axis at x = 0, makes contact, and then climbs away.
\(y = x\) on the other hand has multiplicity 1 for the root at \(x = 0\) so the curve/line crosses through the x-axis at x = 0.

Answer 23

alrighty soooooo lol

Answer 24

So, here's a spot where the question a bit ambiguous in my opinion:

"roots of –3, 5 of multiplicity 2" does that mean roots of
-3, 5, 5
or
-3, -3, 5, 5
Probably it means the first one, but were I doing the problem, I would state outright that that was my interpretation of the problem statement, in case it turned out to be the other that was expected.

Answer 25

So, you need to make a choice: -3, 5, 5 or -3, -3, 5, 5
Then, write out the polynomial:
\[(x+3)(x-5)(x-5) = (x+3)(x-5)^2 = ...\]
or
\[(x+3)(x+3)(x-5)(x-5) = (x+3)^2(x-5)^2 = ...\]

Answer 26

when you expand the polynomial, the highest exponent on x will be the number of product terms you have.