Solved Examples of Evaluating Polynomial Functions

📚 Quick Study Guide

• 🔍 A polynomial function is an expression in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
• 💡 To evaluate a polynomial function at a specific value of $x$, substitute that value into the expression and simplify. For example, to find $f(2)$, replace every $x$ with $2$.
• 📝 Use the order of operations (PEMDAS/BODMAS) to simplify the expression after substitution: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• ➕ Synthetic division can be used as an efficient method for evaluating polynomials, especially when dividing by a linear factor of the form $(x - c)$. The remainder is the value of the polynomial at $x = c$.
• 🔢 When evaluating polynomials, pay close attention to signs, especially when substituting negative values for $x$. Double-check your calculations to avoid errors.

Practice Quiz

1. What is $f(x) = x^2 + 3x - 5$ evaluated at $x = 2$?
1. -3
2. 5
3. 15
4. 20


2. Evaluate $f(x) = -2x^3 + 4x - 1$ at $x = -1$.
1. -7
2. -1
3. 5
4. 1


3. If $f(x) = 3x^4 - 2x^2 + 7$, what is $f(0)$?
1. 0
2. -7
3. 7
4. 8


4. Find $f(-2)$ for $f(x) = x^3 - 4x^2 + x - 6$.
1. -8
2. -16
3. -28
4. 0


5. What is the value of $f(x) = 5x^2 - x + 3$ when $x = -3$?
1. 45
2. 44
3. 51
4. 39


6. Evaluate $f(x) = x^5 - 1$ at $x = 1$.
1. 0
2. 1
3. 2
4. -1


7. Given $f(x) = 2x^3 + x^2 - 5x + 2$, find $f(1)$.
1. 0
2. -2
3. 2
4. 4