Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=x^{4},\) find \(\frac{f(x)-f(3)}{x-3}\)

Short answer

\( (x + 3)(x^2 + 9) \)

Step-by-step solution

- Understand the Function

The given function is \( f(x) = x^4 \). This tells us that for any value of \( x \), \( f(x) \) will be \( x \) raised to the fourth power.

- Evaluate \( f(3) \)

To find \( f(x) - f(3) \), first determine the value of \( f(3) \). Plug \( 3 \) into the function: \[ f(3) = 3^4 = 81 \].

- Set Up the Expression

Now that we have \( f(3) = 81 \), we can set up our expression: \[ \frac{f(x) - f(3)}{x-3} = \frac{x^4 - 81}{x-3} \].

- Factor the Numerator

The numerator \( x^4 - 81 \) can be factored using the difference of squares: \[ x^4 - 81 = (x^2 - 9)(x^2 + 9) \]. Notice that \( x^2 - 9 \) can be further factored: \[ x^2 - 9 = (x - 3)(x + 3) \]. Thus, we have: \[ x^4 - 81 = (x - 3)(x + 3)(x^2 + 9) \].

- Simplify the Fraction

Substitute the factored form back into the fraction: \[ \frac{(x - 3)(x + 3)(x^2 + 9)}{x - 3} \]. Next, cancel out the \( x - 3 \) term from numerator and denominator: \[ \frac{(x - 3)(x + 3)(x^2 + 9)}{x - 3} = (x + 3)(x^2 + 9) \].

- Write the Final Expression

The simplified form of the given expression is: \[ (x + 3)(x^2 + 9) \].

Key concepts

Difference of Squares

The difference of squares is a special type of polynomial factorization. It applies to expressions of the form \[ a^2 - b^2. \] This can be factored into two binomials: \[ (a - b)(a + b). \] In our exercise, we had the expression \[ x^4 - 81. \] Notice that \[ x^4 \] is a perfect square (\[ (x^2)^2 \]) and 81 is a perfect square (\[ 9^2 \]). We recognize this as a difference of squares and factor it as follows: \[ x^4 - 81 = (x^2 - 9)(x^2 + 9). \] To factor further, \[ (x^2 - 9) \] can also be identified as a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3). \] So, the original expression \[ x^4 - 81 \] ultimately becomes: \[ (x - 3)(x + 3)(x^2 + 9). \] Remember, the difference of squares is useful for simplifying complex polynomials.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In our case, we dealt with the fraction \[ \frac{x^4 - 81}{x-3}. \] To simplify rational expressions, we often need to factor the numerator and/or the denominator and then reduce common factors. For our expression, the factoring process gave us: \[ \frac{(x - 3)(x + 3)(x^2 + 9)}{x - 3}. \] By canceling out the common factor of \[ (x - 3) \] in both the numerator and the denominator, we simplified the expression to: \[ (x + 3)(x^2 + 9). \] Important points to remember about rational expressions:

• Always factor completely if possible.
• Check for and cancel out common factors.
• Be aware of restrictions – those values of x that make the denominator zero (for instance, in our expression, \[ x = 3 \] would make the denominator zero and is thus a restriction).

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler 'factor' polynomials that multiply together to give the original polynomial. In our exercise, we factored \[ x^4 - 81. \] Here’s a breakdown of how to factor it step-by-step:

• Identify patterns (like difference of squares).
• Use algebraic identities: For example, \[ a^2 - b^2 = (a - b)(a + b). \] Applied to our polynomial: \[ x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9). \]
• Factor further if possible. Recognize other patterns in the resulting factors: \[ x^2 - 9 = (x - 3)(x + 3). \]

Effective factoring can simplify complex algebraic expressions and solve equations more easily. It’s a fundamental skill in algebra that helps streamline problem-solving processes.