Question

Construct a polynomial funtion f with the following characteristics
zero -3 (multiplicity 2), 2 (multiplicity 1), and 4 (multiplicity 1).
degree 4
contains the point (1,96)

Answer 1

Oh, boy, lots of steps :-)

If we have a degree of 4, we need to have 4 roots, and indeed we have been supplied with 4: x=-3, x=-3, x = 2, x = 4. Do you know how we can construct a polynomial from that information?

Answer 2

nope i don't think i do

Answer 3

Well, if the polynomial = 0 at those spots, that must mean that we can write it as a product of numerous binomials that = 0 at those spots.

(x-a)(x-b)(x-c)(x-d) = 0 when any of those binomials = 0, right?

Answer 4

So, a root at x = -3 means x + 3 = 0, so the binomial for that root will be (x+3). What will the others be?

Answer 5

well a zero at x = -3 will result in a factor (x +3)
x = 2 will result in a factor (x -2)
x = 4 will result in a factor (x - 4)

multiplicity indicates the power of the factor so

multiplicity 2 is

\[(x + 3)^2\]

so the polynomial is

\[P(x) = a(x + 3)^2(x -2)(x-4)\]

you can find the value of a by using the point (1, 96)

substitute x = 1 and P(x) = 96 to get a

hope this helps

Answer 6

(x-2) and (x-4) like what campbell_st said
so would the answer be 2(x+3)^2(x-2)(x-4) since it says the degree is 4 so meaning it would be 2

Answer 7

No, the degree doesn't affect the value of the leading coefficient like that. You need to evaluate the polynomial at x=1 and fudge the leading coefficient so the polynomial equals 96 there.
\[96 = a(1+3)^2(1-2)(1-4)\]

Answer 8

you have the value of a correct.

\[P(x) = 2(x +3)^2(x -2)(x-4)\]

Answer 9

2 is the correct value for a, but it isn't because the degree is 4.

Answer 10

You should also consider whether your assignment includes expanding the polynomial or not.

Answer 11

it doesn't say that it expanding the polynomial. also then what is the degree for?

Answer 12

The degree allows you to check that you've got the proper number of roots. For example, if the degree was 5, you would know that either one of the multiplicities was incorrect, or another root was missing. In this problem, it doesn't really have any function, but if you were coming at this from a different direction, it might be useful. The Fundamental Theorem of algebra states that if the degree is n, you'll have n roots (some of which may be identical, but the sum of the multiplicities over all the roots will be n).

Answer 13

the degree is the highest power or you can add the multiplicity

it will be 4.

Answer 14

You can also use it as a check if you expand the polynomial: with a degree of 4, if your expanded polynomial doesn't have an x^4 term (or has any powers higher than x^4), you've made a mistake somewhere.

Answer 15

oooo okay thank you!