If 3 is a zero of the polynomial f(x), then which of the following is NOT necessarily true?
A. (x – 3) is a factor of the polynomial f(x).
B. 3 is a solution of the equation f(x) = 0.
C. (3, 0) is an x-intercept for f.
D. (0, 3) is a y-intercept for f.
D.
x=0, y = 3 is NOT necessarily true. All you know for sure is that y = 0 when x = 3.
To answer this question, we need to understand the relationship between zeros of a polynomial, factors of a polynomial, and intercepts.
A zero of a polynomial is a value of x for which the polynomial evaluates to zero. In this case, we have a zero of 3 for the polynomial f(x).
Now let's consider each statement and see if it is necessarily true or not:
A. (x – 3) is a factor of the polynomial f(x).
If 3 is a zero of the polynomial f(x), then (x - 3) is a factor of f(x). This statement is necessarily true because the factor theorem states that if a number is a zero of a polynomial, then the corresponding linear factor is a factor of the polynomial.
B. 3 is a solution of the equation f(x) = 0.
Since 3 is a zero of f(x), it means that 3 is a solution to the equation f(x) = 0. This statement is necessarily true because a zero of a polynomial is the same as a solution to the equation f(x) = 0.
C. (3, 0) is an x-intercept for f.
An x-intercept is a point where the graph of the function intersects the x-axis. If 3 is a zero of f(x), then it means that the graph of f(x) intersects the x-axis at x = 3. So, (3, 0) is an x-intercept for f. This statement is necessarily true.
D. (0, 3) is a y-intercept for f.
A y-intercept is a point where the graph of the function intersects the y-axis. The value of the y-intercept is f(0) because the y-axis corresponds to x = 0. Since we only know that 3 is a zero of f(x), we don't have enough information to determine the y-intercept. So, this statement is NOT necessarily true.
Therefore, the correct answer is D. (0, 3) is a y-intercept for f.