To find the value of the polynomial [tex]\( p(x) \)[/tex] at [tex]\( x = \sqrt{3} \)[/tex], follow these steps:
1. Substitute [tex]\( x = \sqrt{3} \)[/tex] into the polynomial [tex]\( p(x) = x^2 - 3 \sqrt{3} x + 5 \)[/tex]:
[tex]\[
p(\sqrt{3}) = (\sqrt{3})^2 - 3 \sqrt{3} (\sqrt{3}) + 5
\][/tex]
2. Compute each term:
- The first term is [tex]\((\sqrt{3})^2\)[/tex]:
[tex]\[
(\sqrt{3})^2 = 3
\][/tex]
- The second term is [tex]\( -3 \sqrt{3} (\sqrt{3})\)[/tex]:
[tex]\[
-3 \sqrt{3} \times \sqrt{3} = -3 \times 3 = -9
\][/tex]
- The third term is a constant:
[tex]\[
5
\][/tex]
3. Add the computed terms together:
[tex]\[
p(\sqrt{3}) = 3 - 9 + 5
\][/tex]
4. Simplify the expression:
[tex]\[
p(\sqrt{3}) = 3 - 9 + 5 = -6 + 5 = -1
\][/tex]
So, the value of [tex]\( p(\sqrt{3}) \)[/tex] is [tex]\( -1 \)[/tex].
