Problem 15 - Algebra for College Students

Question

For Problems \(11-20\), find \(f(c)\) either by using synthetic division and the remainder theorem or by evaluating \(f(c)\) directly. $$ f(n)=4 n^{7}+3 \text { and } c=3 $$

Step-by-Step Solution

Answer

The value of \( f(3) \) is 8751.

Step 1: Understand the Problem

We need to find the value of the function \( f(n) = 4n^7 + 3 \) when \( n = c = 3 \). We can solve this by directly substituting \( n = 3 \) into the function.

Step 2: Substitute the Value of c into the Function

Substitute \( n = 3 \) into the function \( f(n) = 4n^7 + 3 \). This gives us \( f(3) = 4(3)^7 + 3 \).

Step 3: Calculate the Power

Calculate \( 3^7 \). We know that \( 3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2187 \).

Step 4: Multiply by 4

Multiply the result of \( 3^7 \), which is 2187, by 4. This gives us \( 4 \times 2187 = 8748 \).

Step 5: Add the Constant

Now, add the constant 3 to the result from step 4. So, \( 8748 + 3 = 8751 \).

Step 6: State the Final Result

Thus, \( f(3) = 8751 \).

Key Concepts

Remainder Theorem

The Remainder Theorem is a useful tool in polynomial algebra. It states that when a polynomial \( f(x) \) is divided by a linear divisor of the form \( (x - c) \), the remainder of this division is exactly \( f(c) \). This concept provides an efficient shortcut by allowing us to find the remainder without performing the entire division. Instead of dividing, you can simply evaluate the polynomial at \( x = c \).This theorem offers significant utility especially in polynomial factorizations and root finding. If \( f(c) = 0 \), then \( x - c \) is a factor of the polynomial, meaning \( c \) is a root. In our problem, while the exercise suggests using the theorem, it isn't necessary for direct evaluation. However, understanding this theorem enriches your problem-solving toolkit.

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial function at a given point. This is simple; for a polynomial function \( f(x) \), you substitute the given number into the function and compute the result. For example, to evaluate \( f(n) = 4n^7 + 3 \) at \( n = 3 \), you replace every occurrence of \( n \) with 3, as seen in the solution.The steps are straightforward:

• First, substitute: replace \( n \) with the specified value, here \( 3 \), in the function \( f(n). \)
• Next, carry out the arithmetic operations: compute any powers, like \( 3^7 \), then proceed with multiplication and addition according to the expression.

In our case, we found the value of \( f(3) \) to be 8751 using these steps efficiently without division.

Function Substitution

Function substitution involves replacing variables in a function with specific numbers to find a specific value of that function. In simpler words, it's about plugging in a number for a variable and calculating the result. Using substitution, you can determine the function's behavior at a particular point.Here's how it works using our example:

• Identify the function and the variable to replace, such as \( f(n) = 4n^7 + 3 \), where \( n \) is the variable.
• Substitute the variable with a number: replace \( n \) with 3, making it \( f(3) = 4(3)^7 + 3 \).
• Lastly, solve the equation by performing the calculations, like finding \( 3^7 \), multiplying, and adding the constant number.

This method is direct and especially useful when evaluating polynomials at specific points.