In order to determine which function has real zeros at [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex], we can follow these steps:
1. Formulating the Problem: We need to find out which function, when evaluated at [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex], yields a result of zero. This implies that for a function [tex]\( f(x) \)[/tex], [tex]\( f(3) = 0 \)[/tex] and [tex]\( f(7) = 0 \)[/tex].
2. Evaluating Each Function:
- For the function [tex]\( f(x) = x^2 + 4x - 21 \)[/tex]:
[tex]\[
f(3) = 3^2 + 4 \cdot 3 - 21 = 9 + 12 - 21 = 0
\][/tex]
[tex]\[
f(7) = 7^2 + 4 \cdot 7 - 21 = 49 + 28 - 21 = 56 \neq 0
\][/tex]
- Thus, this function does not have zeros at [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex].
- For the function [tex]\( f(x) = x^2 - 4x - 21 \)[/tex]:
[tex]\[
f(3) = 3^2 - 4 \cdot 3 - 21 = 9 - 12 - 21 = -24 \neq 0
\][/tex]
[tex]\[
f(7) = 7^2 - 4 \cdot 7 - 21 = 49 - 28 - 21 = 0
\][/tex]
- Thus, this function does not have zeros at [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex].
- For the function [tex]\( f(x) = x^2 - 10x + 21 \)[/tex]:
[tex]\[
f(3) = 3^2 - 10 \cdot 3 + 21 = 9 - 30 + 21 = 0
\][/tex]
[tex]\[
f(7) = 7^2 - 10 \cdot 7 + 21 = 49 - 70 + 21 = 0
\][/tex]
- This function has zeros at both [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex].
- For the function [tex]\( f(x) = x^2 - 10x - 21 \)[/tex]:
[tex]\[
f(3) = 3^2 - 10 \cdot 3 - 21 = 9 - 30 - 21 = -42 \neq 0
\][/tex]
[tex]\[
f(7) = 7^2 - 10 \cdot 7 - 21 = 49 - 70 - 21 = -42 \neq 0
\][/tex]
- Thus, this function does not have zeros at [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex].
3. Conclusion: The function [tex]\( f(x) = x^2 - 10x + 21 \)[/tex] is the only one that has real zeros at both [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex].
So, the function with real zeros at [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex] is:
[tex]\[
f(x) = x^2 - 10x + 21
\][/tex]
