Suppose that a polynomial function of degree 5 with rational coefficients has the given
numbers as zeros. Find the other zero(s).

Ask by Mckenzie Olson. in the United States

Mar 20,2025

Question

Suppose that a polynomial function of degree 5 with rational coefficients has the given numbers as zeros. Find the other zero(s). $$ -\frac{1}{5}, \sqrt{11},-4 i $$

UpStudy AI · Accepted Answer

We are given a polynomial of degree 5 with rational coefficients and the following zeros:
$$
-\frac{1}{5}, \quad \sqrt{11}, \quad -4 i.
$$

Since the polynomial has rational coefficients, any irrational or complex zeros must occur in conjugate pairs.

1. The zero $$\sqrt{11}$$ is an irrational number. Its conjugate is $$-\sqrt{11}$$, so $$-\sqrt{11}$$ must also be a zero.

2. The zero $$-4i$$ is a complex number. Its conjugate is $$4i$$, so $$4i$$ must also be a zero.

Now we list the zeros we have:
$$
-\frac{1}{5},\quad \sqrt{11},\quad -\sqrt{11},\quad -4i,\quad 4i.
$$

These are all 5 zeros, which is consistent with a degree 5 polynomial.

Thus, the missing zeros are:
$$
-\sqrt{11} \quad \text{and} \quad 4i.
$$
The missing zeros are $$-\sqrt{11}$$ and $$4i$$.

Suppose that a polynomial function of degree 5 with rational coefficients has the given
numbers as zeros. Find the other zero(s).

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