Problem 3 - Algebra for College Students

Question

For Problems 1-10, find \(f(c)\) by (a) evaluating \(f(c)\) directly, and (b) using synthetic division and the remainder theorem. $$ f(x)=x^{3}-2 x^{2}+3 x-1 \text { and } c=-1 $$

Step-by-Step Solution

Answer

The value of \(f(-1)\) is \(-7\).

Step 1: Substitute c into the function

To evaluate \(f(c)\) directly, substitute \(c = -1\) into the function \(f(x) = x^3 - 2x^2 + 3x - 1\). Replace every \(x\) with \(-1\) to get \((-1)^3 - 2(-1)^2 + 3(-1) - 1\).

Step 2: Calculate the result of substitution

Perform the arithmetic operations: \((-1)^3 = -1\), \(-2(-1)^2 = -2\), \(3(-1) = -3\), and subtract 1. Combine these to get \(-1 - 2 - 3 - 1 = -7\). Thus, the value of \(f(-1)\) is \(-7\).

Step 3: Setup synthetic division

For synthetic division, use the coefficients of \(f(x) = x^3 - 2x^2 + 3x - 1\), which are \(1, -2, 3, -1\). Write these down, and place the value of \(c = -1\) to the left of them in synthetic division format.

Step 4: Perform synthetic division

Bring down the first coefficient (1). Multiply by \(-1\) and add to the next coefficient: \(1 \times -1 = -1\); add \(-1\) to \(-2\) to get \(-3\). Repeat: multiply \(-3\) by \(-1\) to get \(3\); add to 3 to get \(6\). Then \(6 \times -1 = -6\); add \(-6\) to \(-1\) to get \(-7\).

Step 5: Interpret the result

The last value of synthetic division result is the remainder. Therefore, by the remainder theorem, \(f(-1) = -7\), verifying the direct evaluation result.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, particularly useful when dividing by a linear factor. It simplifies the process and focuses only on the coefficients of the polynomial, reducing the complexity of calculations.
To perform synthetic division for the polynomial \(f(x) = x^3 - 2x^2 + 3x - 1\) and \(c = -1\):

• List the coefficients: \(1, -2, 3, -1\).
• Write \(c = -1\) to the left in a special division bracket format.
• Bring down the leading coefficient (1) as is.
• Multiply this number by \(c\) and add it to the next coefficient. Repeat for all coefficients.

This method results in a sequence of transformations which culminate in the remainder. The remainder notably appears as the last result in the sequence. In this example, after calculations, the remainder is \( -7 \), confirming our result via the Remainder Theorem as well. This shows not only the efficiency but also the reliability of synthetic division in simplifying polynomial calculations.

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that connects polynomial division with polynomial evaluation. It states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \((x - c)\) is simply the value \(f(c)\).
To use this theorem effectively:

• First, identify \(c\) from your divisor \((x-c)\).
• Compute \(f(c)\) using either direct substitution or synthetic division.

In the example with \(f(x) = x^3 - 2x^2 + 3x - 1\) and \(c = -1\), both direct evaluation and synthetic division resulted in \(-7\). According to the Remainder Theorem, this remainder signifies that \(f(-1) = -7\).
This theorem simplifies the process by reducing the need for a full polynomial division when you only require the remainder, making it both a practical and powerful tool in polynomial analysis.

Polynomial Evaluation

Polynomial evaluation involves calculating the output of a polynomial function for a given input value, \(c\). It can be approached in two primary ways: direct substitution and synthetic division.

• Direct substitution: Replace each variable in the polynomial with the numerical value and perform the arithmetic operations.
• Synthetic division: Use a streamlined process focusing only on coefficients to determine the polynomial's value at \(c\) indirectly through calculating the remainder.

For the given function \(f(x) = x^3 - 2x^2 + 3x - 1\) and \(c = -1\), both methods give \(f(-1) = -7\). The direct method outputs a straightforward calculation, whereas synthetic division provides not only \(f(c)\) but also validates it through the Remainder Theorem.
Understanding these methods harmonizes computational efficiency and mathematical theory, helping to make polynomial evaluation an accessible task regardless of the method chosen.