To find a polynomial function of degree 3 with the given numbers as zeros, we start with the zeros: [tex]\(\sqrt{5}\)[/tex], [tex]\(-\sqrt{5}\)[/tex], and [tex]\(2\)[/tex].
When we know the zeros of a polynomial, we can express the polynomial as the product of factors corresponding to these zeros. So, the polynomial will be:
[tex]\[ f(x) = (x - \sqrt{5})(x + \sqrt{5})(x - 2) \][/tex]
Next, we multiply these factors together to put the polynomial in standard form.
1. First, we multiply the factors involving [tex]\(\sqrt{5}\)[/tex]:
[tex]\[ (x - \sqrt{5})(x + \sqrt{5}) \][/tex]
This is a difference of squares, which simplifies to:
[tex]\[ (x - \sqrt{5})(x + \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5 \][/tex]
2. Now, we multiply this result by the remaining factor, which is [tex]\((x - 2)\)[/tex]:
[tex]\[ (x^2 - 5)(x - 2) \][/tex]
We perform the multiplication by distributing each term in [tex]\( (x - 2) \)[/tex] with [tex]\(x^2 - 5\)[/tex]:
[tex]\[ (x^2 - 5)(x - 2) = x^2(x) - x^2(2) - 5(x) + 5(2) \][/tex]
This simplifies to:
[tex]\[ x^3 - 2x^2 - 5x + 10 \][/tex]
Thus, the polynomial function in standard form is:
[tex]\[ f(x) = x^3 - 2x^2 - 5x + 10 \][/tex]
So, the polynomial function is:
[tex]\[ f(x) = x^3 - 2x^2 - 5x + 10 \][/tex]
