Solved Examples: Defining Polynomial Functions

📚 Quick Study Guide

• 🔍 A polynomial function is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $n$ is a non-negative integer and the coefficients $a_i$ are constants.
• 📈 The degree of a polynomial is the highest power of $x$ in the polynomial. For example, in $f(x) = 3x^4 + 2x^2 - x + 5$, the degree is 4.
• 🔢 The leading coefficient is the coefficient of the term with the highest power of $x$. In the example above, the leading coefficient is 3.
• 🌱 A polynomial function's domain is all real numbers.
• 💡 Zeros of a polynomial are the values of $x$ for which $f(x) = 0$. These are also called roots.

Practice Quiz

1. Question 1: Which of the following is a polynomial function?
   1. A. $f(x) = \sqrt{x} + 1$
   2. B. $f(x) = x^3 - 2x + 1$
   3. C. $f(x) = \frac{1}{x}$
   4. D. $f(x) = |x|$
2. Question 2: What is the degree of the polynomial function $f(x) = 5x^4 - 3x^2 + x - 7$?
   1. A. 2
   2. B. 3
   3. C. 4
   4. D. 5
3. Question 3: What is the leading coefficient of the polynomial function $f(x) = -2x^5 + 4x^3 - x + 9$?
   1. A. 9
   2. B. 4
   3. C. -1
   4. D. -2
4. Question 4: Which of the following statements is true about the domain of a polynomial function?
   1. A. It can only be positive real numbers.
   2. B. It can only be negative real numbers.
   3. C. It is all real numbers.
   4. D. It depends on the degree of the polynomial.
5. Question 5: How many roots does the polynomial $f(x) = (x-1)(x+2)(x-3)$ have?
   1. A. 1
   2. B. 2
   3. C. 3
   4. D. 0
6. Question 6: What is the constant term of the polynomial $f(x) = 7x^3 - 2x + 5$?
   1. A. 7
   2. B. -2
   3. C. 3
   4. D. 5
7. Question 7: Which of the following is NOT a polynomial?
   1. A. $5x^4 + 3x^2 - 2x + 1$
   2. B. $x^2 + 2x - 3$
   3. C. $\frac{1}{x} + x$
   4. D. $7$