Question

$$\text { If } f(x)=x^{2}+5 x, \text { find and simplify } \frac{f(x)-f(c)}{x-c}, x \neq c$$

Short answer

The simplified expression is \( x + c + 5 \).

Step-by-step solution

Understand the given function

The function given is \( f(x) = x^2 + 5x \).

Compute \( f(c) \)

Replace \( x \) with \( c \) in the function to find \( f(c) \): \( f(c) = c^2 + 5c \).

Set up the expression

Set up the expression for \( \frac{f(x) - f(c)}{x - c} \) using the values of \( f(x) \) and \( f(c) \): \[ \frac{f(x) - f(c)}{x - c} = \frac{x^2 + 5x - (c^2 + 5c)}{x - c} \].

Simplify the numerator

Simplify the numerator by distributing the negative sign and combining like terms: \[ x^2 + 5x - c^2 - 5c \]. This simplifies to \[ (x^2 - c^2) + 5(x - c) \].

Factor the numerator

Factor the numerator: \[ (x - c)(x + c) + 5(x - c) \]. Factor \( x - c \) from both terms: \[ (x - c)(x + c + 5) \].

Simplify the entire expression

Cancel \( x - c \) from the numerator and the denominator: \[ \frac{(x - c)(x + c + 5)}{x - c} = x + c + 5 \].

Key concepts

Difference Quotient

The difference quotient is a formula used in calculus to find the rate at which a function changes. It's essentially the foundation of the derivative. For a function given by \( f(x) \), the difference quotient is expressed as:
\[ \frac{f(x+h) - f(x)}{h} \]
In this case, we're working with two distinct points, so we use:
\[ \frac{f(x) - f(c)}{x - c} \]
This helps us find the slope of the secant line between these two points on the graph of the function. Understanding and mastering the difference quotient is crucial in understanding derivatives and the overall concept of differentiation in calculus.

Polynomial Functions

Polynomial functions are algebraic expressions that include terms in the form \( ax^n \), where \( n \) is a non-negative integer. The general form of a polynomial function is:
\[ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]
In simpler terms, it's a sum of multiple terms, each consisting of a variable raised to an integer power and multiplied by a coefficient. For example, the function in our problem is:
\( f(x) = x^2 + 5x \)
This is a polynomial of degree 2, since the highest power of \( x \) is 2. Polynomial functions are continuous and smooth, making them easier to work with compared to other types of functions.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their most basic form. This includes combining like terms, factoring, and canceling common factors. Let's review the steps we took to simplify our given expression:

• First, we substituted \( c \) into the polynomial to get \( f(c) = c^2 + 5c \).
• Next, we set up the difference quotient \( \frac{f(x) - f(c)}{x - c} = \frac{x^2 + 5x - (c^2 + 5c)}{x - c} \).
• We then simplified the numerator: \( x^2 + 5x - c^2 - 5c \) becomes \( (x^2 - c^2) + 5(x - c) \).
• We factored this as \( (x - c)(x + c + 5) \) and canceled \( x - c \) in the numerator and denominator.
• Finally, we are left with the simplified form: \( x + c + 5 \).

Simplifying such expressions makes it much easier to work with them in further calculations and helps in understanding the overall behavior of the function.