Question

The polynomial of degree 5,

P(x) has leading coefficient 1, has roots of multiplicity 2 at x = 1 and x = 0, and a root of multiplicity 1 at
x = -5 Find a possible formula for P(x).

Answer 1

Step-by-step explanation:

Given the information that the polynomial function P(x) has leading coefficient 1, has roots of multiplicity 2 at x = 1 and x = 0, and a root of multiplicity 1 at x = -5, we can use this information to find a possible formula for P(x).

A polynomial function's roots are the values of x that make the function equal to zero. If a root has multiplicity n, it means that (x - root) is a factor of the polynomial function n times.

So, based on the given information, we know that P(x) must be divisible by the factors (x - 1)^2, (x - 0)^2, and (x + 5), and that the leading coefficient must be 1.

Putting it all together, a possible formula for P(x) is:

P(x) = (x - 1)^2 (x - 0)^2 (x + 5) = (x^2 - 2x + 1)(x^2)(x + 5) = x^6 - 7x^5 + 17x^4 - 22x^3 + 17x^2 - 7x + 5

So, the possible formula for P(x) is P(x) = x^6 - 7x^5 + 17x^4 - 22x^3 + 17x^2 - 7x + 5.

Answer 2

Answer:
`P(\text{x}) = \text{x}^2(\text{x}-1)^2(\text{x}+5)`

Step-by-step explanation:

If k is a root of P(x), then x-k is a factor. The multiplicity is the exponent for that factor.

For instance, the root x = 1 with multiplicity 2 leads to the factor
`(\text{x}-1)^2`

The leading coefficient is the number out front. For example, if the leading coefficient was 8, then we'd have
`P(\text{x}) =8\text{x}^2(\text{x}-1)^2(\text{x}+5)`

However, we change that "8" to a "1" since the leading coefficient is 1 here.

That's how we get to

`P(\text{x}) = 1\text{x}^2(\text{x}-1)^2(\text{x}+5)` aka
`P(\text{x}) = \text{x}^2(\text{x}-1)^2(\text{x}+5)`

Optionally if you were to expand everything out, combine like terms, and simplify then you'd get

`P(\text{x}) = \text{x}^2(\text{x}-1)^2(\text{x}+5) = \text{x}^5 + 3\text{x}^4- 9\text{x}^3+5\text{x}^2`

I don't recommend doing this because you lose the information about the roots along with their corresponding multiplicities. It's better to keep things factored.

You can use a graphing tool like Desmos or GeoGebra to verify the answer.