Problem 55 - Algebra 2

Question

Use synthetic substitution to find \(f(3)\) and \(f(-4)\) for each function. $$ f(x)=4 x^{2}-10 x+5 $$

Step-by-Step Solution

Answer

\(f(3) = 11\) and \(f(-4) = 109\).

Step 1: Set Up for Synthetic Substitution for f(3)

To use synthetic substitution for evaluating \(f(3)\), we start by writing down the coefficients of the polynomial \(4x^2 - 10x + 5\). These are 4, -10, and 5. We set these coefficients in a sequence and place 3 to the left of the sequence for synthetic substitution.

Step 2: Perform Synthetic Substitution for f(3)

Bring down the leading coefficient, 4. Then multiply 3 by 4, and add the result (12) to the next coefficient, -10, resulting in 2. Next, multiply 3 by 2, and add to the next coefficient, 5, resulting in 11. Therefore, \(f(3) = 11\).

Step 3: Set Up for Synthetic Substitution for f(-4)

For \(f(-4)\), repeat the same setup. Use the coefficients 4, -10, and 5 again, placing -4 to the left for synthetic substitution.

Step 4: Perform Synthetic Substitution for f(-4)

Bring down the 4. Then multiply -4 by 4 and add to the next coefficient, -10, resulting in -26. Multiply -4 by -26, and add to the final coefficient, 5, resulting in 109. Therefore, \(f(-4) = 109\).

Key Concepts

Polynomial Evaluation

Evaluating a polynomial involves finding the value of the polynomial function for a specific input value. In this case, we want to determine the values of the function \( f(x) = 4x^2 - 10x + 5 \) when \( x \) equals 3 and -4. This operation is called "polynomial evaluation."

Traditionally, we might directly substitute the value of \( x \) into the equation and solve, but synthetic substitution offers a faster method, especially for polynomials with higher degrees.

By systematically utilizing the coefficients of the polynomial, synthetic substitution provides a streamlined approach to polynomial evaluation, emphasizing its efficiency in calculations. This method can be particularly useful when working with more complex polynomials or when performing multiple evaluations at once.

Coefficients

Coefficients are the numerical factors that multiply the variables in a polynomial. In the polynomial \(4x^2 - 10x + 5\), the coefficients are crucial components: 4, -10, and 5. Each coefficient corresponds to the respective power of \(x\) they are associated with.

Understanding coefficients is essential for synthetic substitution, as these numbers form the foundation of the process. The coefficients are arranged in sequence to facilitate operations like addition and multiplication during substitution.

• The leading coefficient is the first number in an ordered list and corresponds to the largest power of \(x\); here, it is 4.
• Next is -10, which comes from the linear term \(-10x\).
• Finally, the constant term, in this instance 5, acts like a coefficient for \(x^0\), or simply represents an independent constant value in the polynomial.

Being familiar with coefficients is critical for simplifying polynomial expressions and evaluating them efficiently using synthetic methods.

Roots of Polynomials

While evaluating polynomials, one might inquire about the possible values of \(x\) that make the entire polynomial equal zero. These values are known as the "roots" or "zeros" of the polynomial.

Finding the roots of a polynomial provides insights into where the graph of the polynomial will intersect the x-axis. Although synthetic substitution, as used in the exercise, is for evaluating the polynomial at specific values, a related technique called synthetic division is often used for finding these roots when one root is known.

Understanding the roots is essential for analyzing polynomial behaviors, solving polynomial equations, and comprehending areas where the polynomial changes sign. This deepens comprehension of both algebraic structures and their graphical interpretations.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly efficient when dividing by linear factors. While in the context of the exercise we used synthetic substitution, synthetic division remains closely related as it builds upon the same principle.

This method reduces the division algorithm to manipulations involving only the coefficients, bypassing the more cumbersome traditional polynomial division.

To perform synthetic division, organize the coefficients and sequentially calculate each step by utilizing multiplication and addition. The process is visually neat and computationally efficient, making it a favored technique in polynomial algebra.

Synthetic division not only assists in polynomial simplification but also aids in discovering polynomial roots, where known or suspected. Together with substitution, these tools present a robust toolkit for handling polynomial expressions efficiently in various mathematical contexts.