To find the value of [tex]\( k \)[/tex] such that one zero of the polynomial [tex]\( x^2 + 3x + k \)[/tex] is 2, follow these steps:
1. Recall the general form of a polynomial and how zeros of the polynomial work. If [tex]\( r \)[/tex] is a zero of the polynomial [tex]\( f(x) \)[/tex], then [tex]\( f(r) = 0 \)[/tex].
2. Given [tex]\( f(x) = x^2 + 3x + k \)[/tex] and knowing that one zero of this polynomial is 2, substitute [tex]\( x = 2 \)[/tex] into the polynomial and set it equal to zero.
[tex]\[
f(2) = (2)^2 + 3(2) + k = 0
\][/tex]
3. Simplify the equation:
[tex]\[
4 + 6 + k = 0
\][/tex]
4. Combine like terms:
[tex]\[
10 + k = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex] by isolating it on one side of the equation:
[tex]\[
k = -10
\][/tex]
Thus, the value of [tex]\( k \)[/tex] is [tex]\( -10 \)[/tex].
