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Magazine Chapter 6: Problem 2
Use synthetic substitution to find \(f(3)\) and \(f(-4)\) for each function. $$ f(x)=5 x^{4}-6 x^{2}+2 $$
Short Answer
Expert verified
\(f(3) = 353\), \(f(-4) = 1186\).
Step by step solution
01
Identify coefficients
For the polynomial \(f(x) = 5x^4 - 6x^2 + 2\), list the coefficients in order of descending powers of \(x\). The coefficient of \(x^3\) is 0 because it is not present in the polynomial. Therefore, the coefficients are: \([5, 0, -6, 0, 2]\).
02
Synthetic substitution setup for \(f(3)\)
Set up the synthetic substitution process by writing 3 to the left and the coefficients \([5, 0, -6, 0, 2]\) to the right. This setup helps in evaluating \(f(3)\).
03
Perform synthetic substitution to find \(f(3)\)
1. Bring down the first coefficient, 5. 2. Multiply by 3 and add to the next coefficient: \(5 \times 3 + 0 = 15\). 3. Multiply by 3 and add to the next: \(15 \times 3 - 6 = 39\). 4. Multiply by 3 and add to the next: \(39 \times 3 + 0 = 117\). 5. Multiply by 3 and add to the last: \(117 \times 3 + 2 = 353\). Thus, \(f(3) = 353\).
04
Synthetic substitution setup for \(f(-4)\)
Now set up the synthetic substitution process by placing \(-4\) to the left and the coefficients \([5, 0, -6, 0, 2]\) to the right. This helps in evaluating \(f(-4)\).
05
Perform synthetic substitution to find \(f(-4)\)
1. Bring down the first coefficient, 5.2. Multiply by -4 and add to the next coefficient: \(5 \times -4 + 0 = -20\).3. Multiply by -4 and add to the next: \(-20 \times -4 - 6 = 74\).4. Multiply by -4 and add to the next: \(74 \times -4 + 0 = -296\).5. Multiply by -4 and add to the last: \(-296 \times -4 + 2 = 1186\).Thus, \(f(-4) = 1186\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomials
Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They can be simple, like a single monomial, or more complex with multiple terms:
- Each term is composed of a coefficient and a power of the variable.
- The degree of the polynomial is determined by the highest power of the variable.
Coefficients
Coefficients are the numbers in front of the variables in each term of a polynomial. They play a crucial role in determining the polynomial's graph and its behavior. In the polynomial \(f(x) = 5x^4 - 6x^2 + 2\):
- The coefficient of \(x^4\) is 5.
- The coefficient of \(x^2\) is -6.
- The constant term is 2, which is also a coefficient.
- Notice the zero coefficients for the absent \(x^3\) and \(x\) terms.
Function Evaluation
Function evaluation involves calculating the value of a function for specific input values. Synthetic substitution is a streamlined method for evaluating such functions, making it easier than substitution by hand.For example, to evaluate \(f(x) = 5x^4 - 6x^2 + 2\) at \(x = 3\):
- Use synthetic substitution with the coefficients \([5, 0, -6, 0, 2]\).
- The result is derived through consecutive operations, finally obtaining \(f(3) = 353\).
Algebra 2
Algebra 2 expands on foundational algebra concepts and introduces more complex functions and equations.A key topic in Algebra 2 is polynomials and their properties, such as:
- Synthetic division and synthetic substitution.
- Finding zeros and graphing polynomial functions.
