Problem 2 - Algebra 2

Question

Use synthetic substitution to find \(f(3)\) and \(f(-4)\) for each function. $$ f(x)=5 x^{4}-6 x^{2}+2 $$

Step-by-Step Solution

Answer

\(f(3) = 353\), \(f(-4) = 1186\).

Step 1: 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: \([5, 0, -6, 0, 2]\).

Step 2: 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 \(f(3)\).

Step 3: 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, \(f(3) = 353\).

Step 4: 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 \(f(-4)\).

Step 5: 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, \(f(-4) = 1186\).

Key Concepts

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.

For example, in the polynomial \(f(x) = 5x^4 - 6x^2 + 2\), there are three terms: \(5x^4\), \(-6x^2\), and a constant term, 2. Polynomials are foundational in Algebra, as they represent equations and functions that are essential in higher-level mathematics.

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.

Understanding coefficients is vital in synthetic substitution, where each coefficient is used systematically to evaluate the function at given points.

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 \(f(3) = 353\).

This approach is efficient, especially for polynomials, as it reduces the need for repeated multiplications and additions, simplifying complex calculations.

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.

The process of synthetic substitution, as demonstrated in evaluating \(f(3)\) and \(f(-4)\) for a polynomial function, is a technique that students need to master.Algebra 2 allows students to explore a variety of algebraic concepts, such as quadratic equations, exponential functions, and logarithms, building valuable skills for higher mathematics.