Find a polynomial function of degree 3 with the given numbers as zeros. Assume that the leading coefficient is 1 . $$ \sqrt{3},-\sqrt{3}, 4 $$

Question

UpStudy AI · Accepted Answer

Let the polynomial be
$$
P(x) = (x-\sqrt{3})(x+\sqrt{3})(x-4).
$$

**Step 1: Multiply the conjugate factors.**

The product of the conjugate factors is:
$$
(x-\sqrt{3})(x+\sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3.
$$

**Step 2: Multiply the quadratic by the remaining factor.**

Now, multiply:
$$
P(x) = (x^2 - 3)(x-4).
$$

Expanding, we have:
$$
P(x) = x^2 \cdot x - x^2 \cdot 4 - 3 \cdot x + 3 \cdot 4.
$$
$$
P(x) = x^3 - 4x^2 - 3x + 12.
$$

Thus, the polynomial is:
$$
\boxed{P(x) = x^3 - 4x^2 - 3x + 12.}
$$
The polynomial is $$ P(x) = x^3 - 4x^2 - 3x + 12 $$.

Find a polynomial function of degree 3 with the given numbers as zeros. Assume that the leading coefficient is 1 . \[ \sqrt{3},-\sqrt{3}, 4 \]