To determine whether [tex]\(\sqrt{3}\)[/tex] and [tex]\(-\sqrt{3}\)[/tex] are zeroes of the polynomial [tex]\(x^2 - 3\)[/tex], we need to evaluate the polynomial at these points and see if the result is equal to zero.
1. Evaluate the polynomial at [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
P(\sqrt{3}) = (\sqrt{3})^2 - 3
\][/tex]
[tex]\[
P(\sqrt{3}) = 3 - 3 = 0
\][/tex]
2. Evaluate the polynomial at [tex]\(-\sqrt{3}\)[/tex]:
[tex]\[
P(-\sqrt{3}) = (-\sqrt{3})^2 - 3
\][/tex]
[tex]\[
P(-\sqrt{3}) = 3 - 3 = 0
\][/tex]
However, when we perform the actual calculations, we find:
[tex]\[
P(\sqrt{3}) \approx -4.440892098500626e-16
\][/tex]
[tex]\[
P(-\sqrt{3}) \approx -4.440892098500626e-16
\][/tex]
These values, [tex]\(-4.440892098500626e-16\)[/tex], are extremely close to zero but not exactly zero due to numerical precision issues.
3. Verify if these values are indeed zero:
- For [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
P(\sqrt{3}) \neq 0
\][/tex]
- For [tex]\(-\sqrt{3}\)[/tex]:
[tex]\[
P(-\sqrt{3}) \neq 0
\][/tex]
Thus, neither [tex]\(\sqrt{3}\)[/tex] nor [tex]\(-\sqrt{3}\)[/tex] are exact zeroes of the polynomial [tex]\(x^2 - 3\)[/tex] given the numerical precision observed.
