Problem 63 - College Algebra

Question

For the following exercises, use synthetic division to determine the quotient involving a complex number. $$ \frac{x^{3}+1}{x-i} $$

Step-by-Step Solution

Answer

Quotient: \(x^2 + ix - 1\), Remainder: \(1 - i\).

Step 1: Identify the Divisor and Format the Division

The divisor in this problem is \(x - i\). We will use synthetic division, where the divisor \(x - i\) corresponds to \(i\) when using synthetic division notation. Format the division by writing \(i\) on the top left side and the coefficients of the dividend \(x^3 + 0x^2 + 0x + 1\) in a row: \([1, 0, 0, 1]\).

Step 2: Bring Down the Leading Coefficient

Start synthetic division by bringing down the leading coefficient (1) to the bottom row. This will be our first entry. It remains as 1.

Step 3: Multiply and Add Iteratively

Next, multiply the leading coefficient (now at the bottom row) by the divisor \(i\) and place the result under the next coefficient. Continue this process step by step:1. Multiply: \(1 \times i = i\).2. Add: \(0 + i = i\).3. Multiply: \(i \times i = i^2 = -1\) (since \(i^2 = -1\)).4. Add: \(0 - 1 = -1\).5. Multiply: \(-1 \times i = -i\).6. Add: \(1 - i\).

Step 4: Write Down the Quotient and Remainder

The bottom row now gives the coefficients of the quotient polynomial, and the last entry is the remainder. The quotient polynomial is \(x^2 + ix - 1\), and the remainder is \(1 - i\).

Step 5: Write the Final Result

The final result of the division is \(x^2 + ix - 1 + \frac{1-i}{x-i}\). This expresses the dividend \(x^3 + 1\) as a combination of the quotient polynomial and the remainder.

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers, introducing an additional unit called the imaginary unit, denoted by \(i\). The defining property of \(i\) is that \(i^2 = -1\). This allows us to solve equations that have no solutions in the real number system, such as \(x^2 + 1 = 0\).
Complex numbers are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. In this expression:

• \(a\) is the real part.
• \(b\) is the imaginary part.

Handling complex numbers involves operations similar to real numbers, but with special rules for dealing with the imaginary unit \(i\). For instance:- Adding and subtracting: add or subtract the real parts separately from the imaginary parts.- Multiplying: use the distributive property, keeping in mind that \(i^2 = -1\).Complex numbers are foundational in many areas of mathematics and physics, as they extend the number system to allow for a broader range of equations and transformations.

Polynomial Division

Polynomial division is a process similar to long division with numbers, but applied to polynomial functions. It's used to divide one polynomial by another, yielding a quotient polynomial and possibly a remainder.
One method for dividing polynomials is synthetic division. This technique simplifies the process when the divisor is in the form \(x - c\). Here are the basic steps involved:

• Identify the divisor, \(x - c\), and use \(c\) in the synthetic division setup.
• List the coefficients of the dividend polynomial.
• Perform iterative steps of multiplying and adding using synthetic division rules.

Synthetic division is particularly useful when the divisor is linear, as it streamlines calculations that would otherwise be complex and time-consuming.

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra related to polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder of this division is \(f(c)\).
The theorem offers a quick way to evaluate the remainder without performing complete division. To use the Remainder Theorem, substitute \(c\) into the polynomial and calculate the result. If you arrive at zero, \(x - c\) is a factor of the polynomial.- It's a powerful tool in polynomial factorization.- Avoids lengthy manual division for finding remainders.In the exercise, the remainder was found using synthetic division, and matched with \(f(i)\), demonstrating this theorem in practice.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two, usually written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients. Quadratic polynomials present parabolic graphs and are fundamental to many aspects of mathematics.
Key features of quadratic polynomials include:

• They can be factored, solved by completing the square, or tackled using the quadratic formula.
• Their graphs are parabolas which can open upwards or downwards depending on the coefficient \(a\).

When you perform synthetic division, as in the exercise, your quotient might end up being a quadratic polynomial if your original polynomial is cubic. The quotient from synthetic division effectively reduces the polynomial's degree, allowing more straightforward analysis and solving of the resulting quadratic expression.