Write a polynomial function of degree 5 with zeros -1, 2, and i. Justify your answer.

Answer 1

You are given roots at -1 and 2, so the first two factors are (x + 1)(x - 2)

If there is a factor of (x - i), then the conjugate must also be a factor. So now we have:

(x + 1)(x - 2)(x - i)(x + i)

Multiply it out:

(x² - x - 2)(x² + 1) = x⁴ - x³ - x² - x - 2

To make it 5th degree, multiply by any factor of the form Ax + B. The easiest one is to simply multiply by x

Answer 2

First step would be to set x equal to all the zeros of the function

x = -1

x = 2

x = i or rather x = sqrt(-1)

Then move the functions over so that they equal 0

x + 1 = 0

x - 2 = 0

For the third square both sides then move over the -1

x = sqrt(-1)

x^2 = -1

x^2 + 1 = 0

Next step is to increase the multiplicity of one of the roots once to increase the degree of the function without adding another zero

(x + 1)^2 (x - 2)(x^2 + 1) = 0

Now exand the first term

(x^2 + 2x + 2)(x - 2)(x^2 + 1) = 0

Next distribute the Second function onto the first

(x^3 + 2x^2 + 2x - 2x^2 - 4x - 4)(x^2 + 1) = 0

Simplify

(x^3 - 2x - 4)(x^2 + 1) = 0

Distribute again

x^5 - 2x^3 - 4x^2 + x^3 - 2x - 4 = 0

Combine like terms and reorder

x^5 - x^3 - 4x^2 - 2x - 4 = 0

Answer 3

So, this will feel backward compared to your normal process of being given a polynomial and finding the zeros.

x = -1 ⇒ x + 1 = 0 ⇒ (x + 1) is the corresponding factor to a zero of -1

x = 2 ⇒ x - 2 = 0 ⇒ (x - 2) is the corresponding factor to a zero of 2

Complex zeros always come in pairs, so if i is a zero, then we know -i is also a zero. Note that ±i = ±√-1

x = ±i ⇒ x = ±√-1 *square each side* ⇒ x² = -1 ⇒ x² + 1 = 0 ⇒ (x² + 1) is the corresponding factor to a zero of ±i

So our current polynomial has the following factored form: f(x) = (x + 1)(x - 2)(x² + 1) this is a multiplicity of degree 4 because of the highest number of x's that would multiply together is 4 (Think leading term from each factor multiplies: x • x • x² = x⁴ )

So to get it to be degree 5 I suggest increasing the multiplicity of one of the first two factors by 1. So potential answers are:

f(x) = (x + 1)²(x - 2)(x² + 1) or f(x) = (x + 1)(x - 2)²(x² + 1)

I hope I made this process clear enough! Feel free to reach out if you need any more help. :)

Shannon L.