Final answer:
The function 5(x) has three distinct real zeros, a repeated zero at x=4 from the perfect square factor, and two additional distinct real zeros from the quadratic factor with a positive discriminant, indicating no complex zeros.
Step-by-step explanation:
To find the zeros of the function 5(x) = (x - 4)²(x² - 7x + 10), we need to examine each factor separately. The first factor, (x - 4)², is a perfect square which indicates it has a repeated zero at x = 4. Considering this is a square, the graph of the function only touches the x-axis at this point and does not cross it, meaning x=4 is a zero of multiplicity 2.
For the second factor, (x² - 7x + 10), we can use the quadratic formula to determine the zeros. The quadratic formula is given by x = [-b ± sqrt(b² - 4ac)]/(2a), for a quadratic equation of the form ax² + bx + c = 0. In this case, a=1, b=-7, and c=10. Calculating the discriminant (b² - 4ac), we find that it is positive (49 - 40 = 9), which means there are two distinct real solutions to this quadratic equation. Thus, the quadratic factor yields two distinct real zeros.
Therefore, the correct statement regarding the zeros of the function is that it has three distinct real zeros: a double zero at x=4 and two additional distinct zeros from the quadratic factor. No complex zeros are present as the discriminant was positive.