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To evaluate a polynomial function at a given value, you use the **substitution method**. This means you replace every instance of the variable, often denoted as \( x \), with the number specified in the problem. For example, to find \( f(-1) \) for the polynomial \( f(x) = x^2 - 3x + 4 \), substitute \( x \) with -1:

\( f(-1) = (-1)^2 - 3(-1) + 4 \).

Doing this substitution correctly is crucial to solving the problem accurately. Make sure to follow the mathematical order of operations (PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction). It will guide you in performing operations one step at a time.

For instance, after substituting \(x = -1\), the next step involves computing each part of the expression as shown.

After substituting the values into the polynomial, the next step is to **simplify the expression**. This involves performing all the operations on the substituted values.

• First, handle any exponents.
• Second, perform any multiplications.
• Finally, add or subtract the remaining terms.

Using our example with \( f(x) = x^2 - 3x + 4 \) and \( f(-1) \):

First, we calculate the exponent:
\( (-1)^2 = 1 \).

Then, we handle the multiplication:
\( -3(-1) = 3 \).

Finally, we perform the addition:
\( 1 + 3 + 4 = 8 \).

You've now simplified and evaluated \( f(-1) \) to be 8.
Simplifying expressions helps in breaking down complexities, making sure every calculation is carried out systematically to avoid mistakes.

A **polynomial** is a mathematical expression consisting of variables, coefficients, and exponents, combined through addition, subtraction, and multiplication. The given function \( f(x) = x^2 - 3x + 4 \) is a polynomial.

Important features of polynomials to note:

• Degree: The highest power of the variable in the polynomial, which in this case is 2.
• Coefficients: The numbers multiplying each term, like -3 in -3x.
• Constant term: A term with no variable, such as 4 in our example.

Polynomials can be evaluated at any point by substituting and simplifying as shown. It's a fundamental skill in algebra that leads to a better understanding of more complex functions.