Let's consider the given polynomial [tex]\(2x^2 + 9x - 5\)[/tex] and the operation performed by David.
When David divided the polynomial [tex]\(2x^2 + 9x - 5\)[/tex] by [tex]\(2x - 1\)[/tex], he was left with a remainder of zero. This means that [tex]\(2x - 1\)[/tex] is a factor of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
To determine which of the given options must be true, we need to find the roots of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
The roots of a polynomial [tex]\(P(x)\)[/tex] are the values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(P(x) = 0\)[/tex].
Given the polynomial [tex]\(2x^2 + 9x - 5\)[/tex], the roots are the solutions to the equation:
[tex]\[
2x^2 + 9x - 5 = 0
\][/tex]
The roots of this polynomial are:
[tex]\[
x = -5 \quad \text{and} \quad x = \frac{1}{2}
\][/tex]
Now, let's evaluate each of the statements based on the roots we found:
A. [tex]\(\frac{1}{2}\)[/tex] must be a root of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
This is true because [tex]\(\frac{1}{2}\)[/tex] is one of the roots we found.
B. 5 must be a root of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
This is false because the roots we found are [tex]\(-5\)[/tex] and [tex]\(\frac{1}{2}\)[/tex], not 5.
C. [tex]\(\frac{-1}{2}\)[/tex] must be a root of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
This is false because the roots we found are [tex]\(-5\)[/tex] and [tex]\(\frac{1}{2}\)[/tex], not [tex]\(\frac{-1}{2}\)[/tex].
D. 0 must be a root of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
This is false because the roots we found are [tex]\(-5\)[/tex] and [tex]\(\frac{1}{2}\)[/tex], not 0.
Therefore, the correct and true statement is:
A. [tex]\(\frac{1}{2}\)[/tex] must be a root of the polynomial [tex]\(2x^2 + 9x - 5\)[/tex].
