91Ó°ÊÓ

Polynomial functions are mathematical expressions that contain variables raised to whole number powers and have coefficients. These are expressed in the form of \(a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\), where each \(a\) represents coefficients and \(n\) is a non-negative integer. For example, in the polynomial function \(f(x) = x^2 - 3x + 2\), you can observe that it includes terms like \(x^2\) and \(-3x\) with coefficients 1 and -3 respectively, and a constant term 2.
Key characteristics of polynomial functions:

• They are continuous and smooth, meaning no sharp corners or breaks.
• The degree of the polynomial is the highest power of the variable. Here, the degree is 2 since the highest exponent of \(x\) is 2.
• Coefficients can be any real number.

Understanding polynomial functions allows you to evaluate them at specific points, such as given in the example where \(x\) was substituted with -7.

Polynomial addition involves combining two or more polynomials into a single polynomial by adding their corresponding coefficients. The operation follows basic arithmetic principles. For instance, if you have \(f(x) = x^2 - 3x + 2\) and \(g(x) = 2x - 4\), to add these polynomial functions, line up like terms and add the coefficients.

Steps in Polynomial Addition

- Align the polynomials according to their degrees.
- Add coefficients of like terms: the constant terms, the coefficients of \(x\), and any higher degree terms separate.After adding \(f(x)\) and \(g(x)\), the resulting polynomial is \[f(x) + g(x) = (x^2 - 3x + 2) + (2x - 4) = x^2 - x - 2.\] Notice how we've summed the coefficients of like terms.

• \(x^2\): Remains the same as there is no corresponding term in \(g(x)\).
• \(-3x\) and \(2x\): Combine to get \(-x\).
• Constant terms \(2\) and \(-4\): Combine to give \(-2\).

This makes polynomial addition systematic, leveraging aligning like terms.

The substitution method is a basic technique used to evaluate functions. It simply means replacing the variable in a function with a given number to find the function's value at that point. This method comes in handy especially in tasks like function evaluation, allowing you to simplify complex algebraic expressions.

How to Use the Substitution Method

- Identify the function(s) to be evaluated. For example, \(f(x) = x^2 - 3x + 2\) and \(g(x) = 2x - 4\).
- Substitute the given value into each function. Let's take \(x = -7\).
- Calculate each function separately:

• For \(f(-7)\), substitute to get \((-7)^2 - 3(-7) + 2 = 72\).
• For \(g(-7)\), substitute to obtain \(2(-7) - 4 = -18\).

- Finally, perform any additional operations needed, such as adding the results from \(f(-7)\) and \(g(-7)\), giving you the result of the new function or expression.This method provides a clear path to arrive at specific numerical results from algebraic expressions, making calculations more straightforward.