To determine the validity of the statement "The function [tex]\( f(x) = x^3 + 3x^2 - 4x + 4 \)[/tex] has four real zeros," let's analyze the polynomial without graphing.
1. Degree of the Polynomial:
- The given function is [tex]\( f(x) = x^3 + 3x^2 - 4x + 4 \)[/tex].
- This is a cubic polynomial since the highest power of [tex]\( x \)[/tex] is 3.
- A polynomial of degree [tex]\( n \)[/tex] (where [tex]\( n \)[/tex] is the highest exponent) can have at most [tex]\( n \)[/tex] real zeros. So, a cubic polynomial can have at most 3 real zeros.
2. Real Zeros:
- We assessed the polynomial to find its real zeros. After solving, it turns out that there is only one real zero for this polynomial.
3. Truth of the Statement:
- The statement claims the polynomial has four real zeros.
- We have established that this polynomial can have at most 3 real zeros and, in fact, it has only one real zero.
- Therefore, the statement is false.
4. Explanation of the Options:
- Option A: "The statement is false because the polynomial function of degree [tex]\( n \)[/tex] has at most [tex]\( n+1 \)[/tex] real zeros." This is incorrect because the maximum number of real zeros for a polynomial of degree [tex]\( n \)[/tex] is [tex]\( n \)[/tex], not [tex]\( n+1 \)[/tex].
- Option B: This option is incorrect because it states the maximum number of real zeros for a degree [tex]\( n \)[/tex] polynomial is [tex]\( n+1 \)[/tex] and that the polynomial has 4 real zeros.
- Option C: This option is incorrect because it states that the polynomial has 4 real zeros, which we determined is false.
- Option D: "The statement is false because [tex]\( f(x) = x^3 + 3x^2 - 4x + 4 \)[/tex], [tex]\( n \)[/tex] is 3." This is true because the polynomial can have at most 3 real zeros and not 4.
The correct answer is:
D. The statement is false because [tex]\( f(x) = x^3 + 3x^2 - 4x + 4 \)[/tex], [tex]\( n \)[/tex] is 3.
