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Answer :
Since the polynomial is required to have integer coefficients and a leading coefficient of 1, the irrational root [tex]$\sqrt{3}$[/tex] must appear with its conjugate, [tex]$-\sqrt{3}$[/tex]. Therefore, the polynomial must have the four roots
[tex]$$
\sqrt{3},\quad -\sqrt{3},\quad 4,\quad -4.
$$[/tex]
A polynomial with these roots can be written as the product of its corresponding linear factors:
[tex]$$
f(x) = (x-\sqrt{3})(x+\sqrt{3})(x-4)(x+4).
$$[/tex]
Step 1. First, multiply the factors corresponding to the irrational roots:
[tex]$$
(x-\sqrt{3})(x+\sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3.
$$[/tex]
Step 2. Next, multiply the factors corresponding to the rational roots:
[tex]$$
(x-4)(x+4) = x^2 - 4^2 = x^2 - 16.
$$[/tex]
Step 3. Now, multiply the two quadratic factors:
[tex]$$
f(x) = (x^2-3)(x^2-16).
$$[/tex]
Using the distributive property:
[tex]$$
f(x) = x^2(x^2-16) - 3(x^2-16) = x^4 - 16x^2 - 3x^2 + 48.
$$[/tex]
Simplify by combining like terms:
[tex]$$
f(x) = x^4 - (16+3)x^2 + 48 = x^4 - 19x^2 + 48.
$$[/tex]
Thus, the polynomial function of lowest degree with the specified roots is
[tex]$$
\boxed{x^4 - 19x^2 + 48}.
$$[/tex]
[tex]$$
\sqrt{3},\quad -\sqrt{3},\quad 4,\quad -4.
$$[/tex]
A polynomial with these roots can be written as the product of its corresponding linear factors:
[tex]$$
f(x) = (x-\sqrt{3})(x+\sqrt{3})(x-4)(x+4).
$$[/tex]
Step 1. First, multiply the factors corresponding to the irrational roots:
[tex]$$
(x-\sqrt{3})(x+\sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3.
$$[/tex]
Step 2. Next, multiply the factors corresponding to the rational roots:
[tex]$$
(x-4)(x+4) = x^2 - 4^2 = x^2 - 16.
$$[/tex]
Step 3. Now, multiply the two quadratic factors:
[tex]$$
f(x) = (x^2-3)(x^2-16).
$$[/tex]
Using the distributive property:
[tex]$$
f(x) = x^2(x^2-16) - 3(x^2-16) = x^4 - 16x^2 - 3x^2 + 48.
$$[/tex]
Simplify by combining like terms:
[tex]$$
f(x) = x^4 - (16+3)x^2 + 48 = x^4 - 19x^2 + 48.
$$[/tex]
Thus, the polynomial function of lowest degree with the specified roots is
[tex]$$
\boxed{x^4 - 19x^2 + 48}.
$$[/tex]
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