Problem 57 - Algebra for College Students

Question

The remainder and factor theorems are true for any complex value of \(c\). Therefore, for Problems \(56-58\), find \(f(c)\) by (a) using synthetic division and the remainder theorem, and (b) evaluating \(f(c)\) directly. $$ f(x)=x^{2}+4 x-2 \text { and } c=1+i $$

Step-by-Step Solution

Answer

The remainder when \(f(x)\) is divided by \(x-(1+i)\) is \(2+6i\), and \(f(1+i) = 2+6i\).

Step 1: Set Up Synthetic Division

To find the remainder of \(f(x) = x^2 + 4x - 2\) for \(c = 1+i\) using synthetic division, write the coefficients of the polynomial as \([1, 4, -2]\) and the number \(c = 1+i\) to the left.

Step 2: Perform Synthetic Division

Position \(1+i\) to the left. Drop the leading coefficient (1) down as it is. Multiply \(1\) by \(1+i\) to get \(1+i\), and place it under the next coefficient (4). Add \(4\) and \(1+i\) to get \(5+i\). Next, multiply \(5+i\) by \(1+i\), which gives \(4+6i\). Add \(-2\) and \(4+6i\) to get \(2+6i\). This last result is the remainder: \(2+6i\).

Step 3: Verify Using the Remainder Theorem

The Remainder Theorem states that the remainder of \(f(x)\) when divided by \((x-c)\) is \(f(c)\). Thus, \(f(1+i) = 2+6i\).

Step 4: Evaluate Directly

To compute \(f(1+i)\) directly, substitute \(1+i\) into \(f(x) = x^2 + 4x - 2\): - First, calculate \((1+i)^2 = 1 + 2i - 1 = 2i\).- Next, compute \(4(1+i) = 4 + 4i\).- Add them: \(f(1+i) = 2i + 4 + 4i - 2 = 2 + 6i\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear factor, and it is especially helpful when dealing with complex numbers. Instead of performing the long polynomial division, synthetic division allows you to work mainly with coefficients and a "synthetic" form of computation.

• Arrange the coefficients from your polynomial into a sequence, e.g., for the polynomial \(f(x) = x^2 + 4x - 2\), you use \([1, 4, -2]\).
• The complex number \(c = 1+i\) is positioned to the left of these coefficients.
• Begin by bringing down the leading coefficient (in our example, it's \(1\)) directly below the line as the starting point.
• Multiply this number by \(c\) and place the result under the next coefficient; then add this result to the next coefficient.
• Continue this process until complete; the final number obtained is your remainder \(f(c)\).

In our exercise, applying these steps results in a remainder of \(2+6i\). Synthetic division is not only concise but also enhances understanding of the polynomial structure, especially when solving for complex roots.

Complex Numbers

Complex numbers extend our number system beyond reals with the inclusion of \(i\), the imaginary unit, where \(i^2 = -1\). They are presented in the form \(a + bi\), where \(a\) and \(b\) are real numbers. In polynomial functions, complex numbers like \(c = 1+i\) can be used as values to evaluate or divide polynomials.

• Complex arithmetic involves operations such as addition, subtraction, multiplication, and division.
• When multiplying complex numbers, such as \((1+i)(5+i)\), apply the distributive property: \[1 \cdot 5 + 1 \cdot i + i \cdot 5 + i^2\].In this case: \[5 + i + 5i - 1 = 4 + 6i\].
• This showcases how real and imaginary components are handled distinctly yet complementarily.

Using complex numbers provides a complete solution capability for polynomial equations, accommodating both real and imaginary results.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial function at a given point. When dealing with polynomials and complex numbers, it becomes imperative to appreciate how substitution affects the evaluation process.

• Substitute the complex number into every occurrence of \(x\), e.g., \(f(x) = x^2 + 4x - 2\) with \(x = 1+i\).
• Compute the powers first, such as \((1+i)^2 \), using \((a+b)^2 = a^2 + 2ab + b^2\), which yields \(2i\) since \(i^2 = -1\).
• Then multiply, \(4(1+i) = 4 + 4i\) and finally, combine all parts, \(2i + 4 + 4i - 2\) simplifying to \(2+6i\).

Direct evaluation requires a good command of operations on complex numbers but provides a clear and thorough way to verify results from methods like synthetic division.