Find a polynomial function of lowest degree
with rational coefficients that has the given
numbers as some of its zeros.

Question

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. $$ -4 i, 7 $$

UpStudy AI · Accepted Answer

Let the zeros be $$ -4i $$ and $$ 7 $$. Since the polynomial must have rational coefficients, the complex roots occur in conjugate pairs, so $$ 4i $$ is also a zero.

The zeros of the polynomial are:
$$
-4i,\quad 4i,\quad 7
$$

The corresponding factors are:
$$
(x + 4i),\quad (x - 4i),\quad (x - 7)
$$

Multiply the factors corresponding to the complex roots:
$$
(x + 4i)(x - 4i) = x^2 - (4i)^2 = x^2 - (-16) = x^2 + 16
$$

Now, multiply by the factor corresponding to $$ 7 $$:
$$
(x - 7)(x^2 + 16)
$$

Expanding the product:
$$
(x - 7)(x^2 + 16) = x\cdot x^2 + x\cdot 16 - 7\cdot x^2 - 7\cdot 16 = x^3 - 7x^2 + 16x - 112
$$

Thus, the polynomial of lowest degree with rational coefficients that has $$ -4i $$ and $$ 7 $$ as zeros is:
$$
x^3 - 7x^2 + 16x - 112
$$
The polynomial is $$ x^3 - 7x^2 + 16x - 112 $$.

Find a polynomial function of lowest degree
with rational coefficients that has the given
numbers as some of its zeros.

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