Let's determine what a root of a polynomial function is step by step.
1. Understanding a Polynomial: A polynomial is an expression consisting of variables (also known as indeterminates) and coefficients, that involves only addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, [tex]\( f(x) = 2x^3 - 3x^2 + x - 5 \)[/tex] is a polynomial.
2. Definition of a Root:
A root of a polynomial function is a specific value of the variable (often [tex]\(x\)[/tex]) such that when substituted into the polynomial, the result is zero. That is, if [tex]\( r \)[/tex] is a root of the polynomial [tex]\( f(x) \)[/tex], then [tex]\( f(r) = 0 \)[/tex].
3. Evaluating the Choices:
- A. A coefficient of the polynomial that is equal to zero: Coefficients are the numbers in front of the variables in a polynomial and don't directly relate to finding where the polynomial equals zero.
- B. The value of the polynomial when zero is substituted for the variable: This typically gives the constant term of the polynomial or the y-intercept when plotted on a graph.
- C. The coefficient of the leading term of the polynomial: The leading term is the term with the highest exponent. This coefficient doesn’t determine where the polynomial equals zero.
- D. A value of the variable that makes the polynomial equal to zero: This is the definition of a root of a polynomial.
4. Conclusion:
After carefully analyzing the options, the correct answer is:
D. A value of the variable that makes the polynomial equal to zero.
Thus, a root of a polynomial function is indeed a value of the variable that makes the polynomial equal to zero.
