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Magazine Chapter 5: Problem 106
Use synthetic division to find the function values. \(f(x)=x^{4}-2 x^{3}+x-6 ;\) find \(f(-1)\)
Short Answer
Expert verified
f(-1) = -4
Step by step solution
01
Set Up Synthetic Division
Write down the coefficients of the polynomial function. For the function \(f(x) = x^4 - 2x^3 + x - 6\), the coefficients are \[1, -2, 0, 1, -6\].The missing term \(x^2\) has a coefficient of 0.
02
Synthetic Division Setup
Place \( -1 \) outside the synthetic division bracket. Write the coefficients \(1, -2, 0, 1, -6\) inside the bracket.
03
Synthetic Division Calculation
1. Bring down the first coefficient (1) to the bottom row.2. Multiply \( -1 \) by 1 and write the result under the second coefficient.3. Add the second coefficient \( -2 \) and the value obtained ( \( -1 \)).This results in \( -3\).4. Continue this process for all coefficients. Here are the steps in detail:1. Write down the 1.2. \( -1 \times 1 = -1 \)3. \( -2 + (-1) = -3 \)4. \( -1 \times -3 = 3 \)5. \( 0 + 3 = 3 \)6. \( -1 \times 3 = -3 \)7. \( 1 + (-3) = -2 \)8. \( -1 \times -2 = 2 \)9. \( -6 + 2 = -4 \)
04
Identify the Result
The last number in the bottom row is the remainder, which in this case is the function value at \( x = -1 \). Thus, \( f(-1) = -4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Function
A polynomial function is a type of mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. For example, the polynomial function given in the exercise is:
\(f(x) = x^4 - 2x^3 + x - 6\)
This function consists of four terms, and each term includes a variable raised to a non-negative power and a coefficient. The function is said to be of the fourth degree because the highest power of the variable (\( x\)) is 4. Here's a quick breakdown of polynomial functions:
\(f(x) = x^4 - 2x^3 + x - 6\)
This function consists of four terms, and each term includes a variable raised to a non-negative power and a coefficient. The function is said to be of the fourth degree because the highest power of the variable (\( x\)) is 4. Here's a quick breakdown of polynomial functions:
- Terms: Individual parts of the polynomial separated by plus or minus signs.
- Coefficient: A numerical factor in each term (e.g., in \( -2x^3\), -2 is the coefficient).
- Degree: The highest power of the variable in the polynomial determines its degree (e.g., 4 in \( x^4\)).
Synthetic Division Steps
Synthetic division is a simplified method of dividing a polynomial by a binomial of the form \(x - c\). It's faster and requires less work compared to traditional long division. Here's a step-by-step guide based on the given exercise:
1. **Set Up Synthetic Division**:
2. **Perform the Division**:
3. **Identify the Result**:
1. **Set Up Synthetic Division**:
- Write down the coefficients of the polynomial \(f(x) = x^4 - 2x^3 + x - 6\). Here, the coefficients are \(1, -2, 0, 1, -6\). The 0 is for the missing term \(x^2\).
- Place \( -1 \) (the value we're evaluating) outside the synthetic division bracket.
2. **Perform the Division**:
- Bring down the first coefficient (1) to the bottom row.
- Multiply \(-1\) by the number in the bottom row and add this result to the next coefficient. Repeat this process for all coefficients.
- Write down 1.
- \( -1 \times 1 = -1 \); add to \( -2 \), resulting in \( -3 \).
- Multiply \( -1 \) by \( -3 \) to get 3; add to the next coefficient (0), resulting in 3.
- Repeat until you reach the end of the coefficients.
3. **Identify the Result**:
- The last number in the bottom row is the remainder, indicating the function evaluation result (\( f(-1) = -4 \)).
Function Evaluation
Function evaluation is the process of determining the output of a function for a specific value of the input variable. In our exercise, we evaluate the function \( f(x) = x^4 - 2x^3 + x - 6 \) at \( x = -1 \). Here's how to break it down:
1. **Substitute the Value**:
2. **Calculate Each Term**:
3. **Final Sum**:
1. **Substitute the Value**:
- Substitute \( x = -1 \) directly into the polynomial function.
2. **Calculate Each Term**:
- Calculate the value of each term in the polynomial for \( x = -1 \).
- \( (-1)^4 = 1 \)
- \( -2(-1)^3 = -2(-1) = 2 \)
- \( (-1) + (-6) = -7 \)
- Combine them to get the final value.
3. **Final Sum**:
- Add all the calculated values: \( 1 + 2 + (-1) + (-6) = -4 \).
