Problem 14 - 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. $$ f(x)=x^{4}-8 x^{3}+9 x^{2}-15 x+2 \text { and } c=7 $$

Step-by-Step Solution

Answer

The value \(f(7)\) is \(-5\).

Step 1: Evaluate Directly using Substitution

To find \( f(c) \), where \( c = 7 \), substitute \( c \) into the function. This means we'll calculate:\[ f(7) = 7^4 - 8 \times 7^3 + 9 \times 7^2 - 15 \times 7 + 2 \]

Step 2: Calculate \( 7^4 \)

First, calculate the highest power of 7. So, \( 7^4 = 7 \times 7 \times 7 \times 7 = 2401 \).

Step 3: Calculate \( 8 \times 7^3 \)

Next, calculate \( 8 \times 7^3 \). First, find \( 7^3 = 7 \times 7 \times 7 = 343 \). Then multiply by 8 to get: \( 8 \times 343 = 2744 \).

Step 4: Calculate \( 9 \times 7^2 \)

Now calculate \( 9 \times 7^2 \). First, find \( 7^2 = 49 \). Then multiply by 9 to get: \( 9 \times 49 = 441 \).

Step 5: Calculate \( 15 \times 7 \)

Calculate \( 15 \times 7 = 105 \).

Step 6: Combine Terms

Now put all calculated parts back into the expression: \[ f(7) = 2401 - 2744 + 441 - 105 + 2 \]

Step 7: Perform Arithmetic Operations

Calculate the above expression step-by-step:1. \( 2401 - 2744 = -343 \)2. \( -343 + 441 = 98 \)3. \( 98 - 105 = -7 \)4. \( -7 + 2 = -5 \)

Step 8: State the Result

The calculated result of \( f(7) \) is \(-5\). Thus, \( f(c) = -5 \) when \( c = 7 \).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, especially useful for dividing a polynomial by a linear factor of the form \( x - c \). It is a speedy way to divide polynomials when the divisor is in this specific format.
This process uses coefficients and is often quicker than long division for polynomials.

To start synthetic division, you use the value of \( c \) (from \( x-c \)) and write down the coefficients of the polynomial in order. You then perform a series of multiplications and additions to get the remainder. This remainder, thanks to the Remainder Theorem, equals \( f(c) \), where \( f \) is the given polynomial.

Here's a simple outline of the steps:

• Write down the coefficients of the polynomial in descending order of power.
• Bring down the leading coefficient as is.
• Multiply \( c \) with the newly written number, write the result underneath the next coefficient, and add these vertically.
• Repeat the multiplication and addition till the end.
• The last value you get is the remainder, which is \( f(c) \).

This makes synthetic division highly efficient for evaluating polynomials at specific points.

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial function at a specific point. This can be done by directly substituting the given value into the polynomial and calculating the result.

In our example, we evaluate the polynomial \( f(x) = x^4 - 8x^3 + 9x^2 - 15x + 2 \) at \( x=7 \). By calculating each term's value and combining them, we determine the function's output at that point.

Direct evaluation is straightforward and involves these steps:

• Substitute the specific value into the polynomial equation.
• Calculate each power of the value, ensuring accuracy in arithmetic operations.
• Multiply the calculated powers by their respective coefficients.
• Add or subtract these results sequentially.

This approach is particularly useful for small polynomials or cases where precision is paramount.

Function Substitution

Function substitution is a critical technique in evaluating polynomial functions, where a specific number is replaced into the function's variable. This process is essentially plugging in values to determine the output.

To evaluate the function \( f(x) = x^4 - 8x^3 + 9x^2 - 15x + 2 \) at \( x=7 \), each term containing \( x \) is replaced by 7:

• The term \( x^4 \) becomes \( 7^4 \).
• Similarly, \( 8x^3 \) changes to \( 8 \times 7^3 \), and so on for all terms.

This substitution allows us to find "real-world" values of functions at given points. It's the most direct method of evaluating polynomials and is crucial for confirming results obtained via methods like synthetic division.

Polynomial Functions

Polynomial functions are mathematical expressions featuring variables raised to whole number powers. They consist of terms like \( ax^n \), where \( a \) represents the coefficient and \( n \) is a non-negative integer.

Such functions are highly versatile and appear in numerous mathematical contexts. They can model various real-world scenarios due to their smooth and predictable nature. Each term in a polynomial function contributes to its overall shape and behavior.

Key features of polynomial functions include:

• Degree: The highest power of the variable, which determines the function's general form.
• Coefficients: Multipliers for each term, influencing the graph's amplitude and shape.
• Constant term: The number added at the end, impacting the function's vertical shift.

Understanding these components helps in visualizing and solving polynomial function problems effectively.