Let's analyze the given function step-by-step to determine the correct statements:
The function is given by:
[tex]\[ k(x) = (3x - 4)(x + 5) \][/tex]
### Step 1: Find the Zeros of the Function
To find the zeros of the function, we set [tex]\( k(x) \)[/tex] equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ (3x - 4)(x + 5) = 0 \][/tex]
This equation will be zero if either factor is zero:
1. [tex]\( 3x - 4 = 0 \)[/tex]
[tex]\[
3x = 4
\][/tex]
[tex]\[
x = \frac{4}{3}
\][/tex]
2. [tex]\( x + 5 = 0 \)[/tex]
[tex]\[
x = -5
\][/tex]
Therefore, the zeros of the function are [tex]\(\frac{4}{3}\)[/tex] and [tex]\(-5\)[/tex].
### Step 2: Analyze the Zeros
Now, we'll check the given conditions:
1. Both zeros are positive:
- The zeros are [tex]\(\frac{4}{3}\)[/tex] and [tex]\(-5\)[/tex].
- One of the zeros ([tex]\(-5\)[/tex]) is negative.
- Hence, this statement is false.
2. Both zeros are integers:
- The zeros are [tex]\(\frac{4}{3}\)[/tex] and [tex]\(-5\)[/tex].
- One of the zeros ([tex]\(\frac{4}{3}\)[/tex]) is not an integer.
- Hence, this statement is false.
3. Both zeros are negative:
- The zeros are [tex]\(\frac{4}{3}\)[/tex] and [tex]\(-5\)[/tex].
- One of the zeros ([tex]\(\frac{4}{3}\)[/tex]) is positive.
- Hence, this statement is false.
4. Both zeros are less than 2:
- The zeros are [tex]\(\frac{4}{3}\)[/tex] and [tex]\(-5\)[/tex].
- Both [tex]\(\frac{4}{3}\)[/tex] (which is approximately 1.33) and [tex]\(-5\)[/tex] are less than 2.
- Hence, this statement is true.
### Conclusion
Based on the analysis above, the correct statement about the function [tex]\( k(x) = (3x - 4)(x + 5) \)[/tex] is:
- The zeros of function [tex]\( k \)[/tex] are both less than 2.
