To determine if the point [tex]\((2, 0)\)[/tex] lies on the graph of the polynomial [tex]\(P(x) = x^4 - 2x^3 - x + 2\)[/tex], we need to evaluate the polynomial at [tex]\(x = 2\)[/tex] and check if the result equals 0.
Here are the steps:
1. Substitute [tex]\(x = 2\)[/tex] into the polynomial [tex]\(P(x)\)[/tex]:
[tex]\[
P(2) = (2)^4 - 2(2)^3 - 2 + 2
\][/tex]
2. Calculate each term individually:
- [tex]\( (2)^4 = 16 \)[/tex]
- [tex]\( 2(2)^3 = 2 \cdot 8 = 16 \)[/tex]
- [tex]\(-2\)[/tex]
3. Combining the results:
[tex]\[
P(2) = 16 - 16 - 2 + 2
\][/tex]
4. Add and subtract the terms:
[tex]\[
P(2) = 16 - 16 = 0; \quad -2 + 2 = 0
\][/tex]
[tex]\[
P(2) = 0
\][/tex]
Since [tex]\(P(2) = 0\)[/tex], we have [tex]\(P(2) = 0\)[/tex], which means the polynomial value at [tex]\(x = 2\)[/tex] is [tex]\(0\)[/tex].
Therefore, the point [tex]\((2, 0)\)[/tex] does indeed lie on the graph of the polynomial [tex]\(P(x)\)[/tex].
The correct answer is:
A. True
