To find a polynomial function of degree 3 with the given numbers as zeros and assuming the leading coefficient is 1, we follow these steps:
### Step 1: Identify the zeros
The zeros of the polynomial are [tex]\( \sqrt{3} \)[/tex], [tex]\( -\sqrt{3} \)[/tex], and [tex]\( 6 \)[/tex].
### Step 2: Write the polynomial in factored form
Since the zeros of the polynomial are [tex]\( \sqrt{3} \)[/tex], [tex]\( -\sqrt{3} \)[/tex], and [tex]\( 6 \)[/tex], the polynomial can be written in its factored form as:
[tex]\[
f(x) = (x - \sqrt{3})(x + \sqrt{3})(x - 6)
\][/tex]
### Step 3: Expand the polynomial
First, we start by multiplying the first two factors, [tex]\((x - \sqrt{3})(x + \sqrt{3})\)[/tex].
Recall the difference of squares formula, [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex]:
[tex]\[
(x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3
\][/tex]
Next, multiply [tex]\((x^2 - 3)\)[/tex] by the remaining factor, [tex]\((x - 6)\)[/tex]:
[tex]\[
(x^2 - 3)(x - 6)
\][/tex]
To expand this, we distribute [tex]\(x^2 - 3\)[/tex] to each term inside the parentheses:
[tex]\[
(x^2 - 3)(x - 6) = x^2(x - 6) - 3(x - 6)
\][/tex]
Now, distribute each term:
[tex]\[
= x^3 - 6x^2 - 3x + 18
\][/tex]
Thus, the polynomial in expanded form is:
[tex]\[
f(x) = x^3 - 6x^2 - 3x + 18
\][/tex]
### Step 4: Write the final polynomial
Combining the expanded terms, the final polynomial function of degree 3 with the given zeros [tex]\( \sqrt{3} \)[/tex], [tex]\( -\sqrt{3} \)[/tex], and [tex]\( 6 \)[/tex], and a leading coefficient of 1, is:
[tex]\[
f(x) = 1.0x^3 - 9.46410161513776x^2 + 23.7846096908265x - 18.0
\][/tex]
This is the polynomial function that satisfies the given conditions.
