91Ó°ÊÓ

Polynomial functions are mathematical expressions that involve sums and differences of terms that consist of variables raised to whole-number exponents.
Polynomials can take many forms but they often look like this: \(ax^n + bx^{n-1} + \text{...} + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are coefficients and \(n\) is a non-negative integer.
In our case, we have a quadratic polynomial function \(f(x) = x^2 - 9\). This means that \(f(x)\) consists of one variable, \(x\), raised to powers which in this case are 2 (the squared term) and 0 (a constant term, -9).
Polynomials are very versatile; they can include many terms or just a few. Remember, the degree of a polynomial is determined by the highest exponent of the variable.

Function subtraction is when you subtract one function from another.
Given two functions, \(f(x)\) and \(h(x)\), function subtraction gives us \((f - h)(x)\).
To perform function subtraction, follow these steps:

• Identify the given functions
• Write the function expression for subtraction. This is typically \(f(x) - h(x)\)
• Substitute the individual functions into this expression
• Simplify the resulting expression (combine like terms)

Let's see this in our given exercise:
We have \(f(x) = x^2 - 9\) and \(h(x) = x - 3\). The function subtraction will then be written as \((f - h)(x)\).
Substituting the given functions:\((f - h)(x) = (x^2 - 9) - (x - 3) = x^2 - 9 - x + 3\).
This simplification step leads us to our simplified expression.

Simplifying expressions means to combine like terms to make the expression as tidy and efficient as possible.
Like terms are terms that have the same variable raised to the same power. You add or subtract their coefficients.
In our example, we simplified the expression \((f - h)(x)\):\((f - h)(x) = x^2 - 9 - x + 3\).
We need to combine the constants -9 and 3:
\(-9 + 3 = -6\). Thus, our simplified expression is \(x^2 - x - 6\).
This process of combining like terms ensures that our final expression is concise and easily understandable.
We use this approach commonly in algebra to make equations more manageable and to solve for the desired variable.