Problem 61 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Quotient is 1 with a remainder of \(1-i\).

Step 1: Understand Synthetic Division Setup

Synthetic division requires expressing the divisor in the form \((x - a)\). In this exercise, our divisor \(x+i\) can be rewritten as \(x-(-i)\), so \(a = -i\). The polynomial to be divided is \(x+1\), which corresponds to the coefficients \([1, 1]\).

Step 2: Set Up Synthetic Division Table

Write down the coefficients of the polynomial \(x+1\) which are \([1, 1]\). Next, place the value \(-i\) (our \(a\) value) to the left of the division bar in the synthetic division table.

Step 3: Perform Synthetic Division Steps

1. Bring down the leading coefficient (1) to the bottom row.2. Multiply the number underneath the division bar (-i) by \(1\), the current bottom number, to get \(-i\).3. Add \(1 + (-i)\) to obtain the result \(1 - i\), and place it in the bottom row.

Step 4: Determine the Quotient

The result in the bottom row represents the quotient. Since we started with a linear polynomial, the last result \(1 - i\) is the constant term of the quotient. The quotient is therefore \(1\), with a remainder of \(1 - i\), meaning our expression is \((1)\) with remainder \(1-i\) over \(x+i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. The imaginary part is represented with 'i', which is the square root of -1. A typical complex number looks like this:

• Real part: a
• Imaginary part: bi

Together, they are expressed as \(a + bi\). Let's break down what each part means:

• Real Part: This is just like any regular number. Examples include 0, -3, or 2.5.
• Imaginary Part: This sounds more complicated than it is. If you imagine multiplying the square root of -1 (which we call 'i') by any number, you get the imaginary part. So, \(3i\) means 3 times the square root of -1.

Complex numbers are useful in engineering, physics, and many areas of mathematics, especially when dealing with polynomial equations that don't have any real solutions.

Polynomial Division

Polynomial division is a process, much like long division with numbers, that allows us to divide one polynomial by another. But it is a bit more abstract:Here's the process in simple steps:

• Identify the divisor and dividend: The dividend is the polynomial you are dividing into and the divisor is the one doing the dividing.
• Use the division algorithm: We alternate between multiplying and subtracting to simplify the division result.

Synthetic division simplifies this process, especially when dividing by a linear polynomial. In our original exercise, the divisor is \(x+i\), and the polynomial being divided is \(x+1\). While usually, polynomials have real numbers, here you get to use complex numbers in the calculations!

Quotient and Remainder

When you divide any polynomial, just like numbers, you find a quotient and sometimes a remainder. Let's understand what they mean:

• Quotient: This is the result you get after dividing one polynomial by another. It shows how many times the divisor fits into the dividend.
• Remainder: Often, there is a leftover part that doesn't perfectly divide. This is your remainder.

In the exercise of dividing \(x+1\) by \(x+i\) using synthetic division, the process gives us a quotient of 1 and a remainder of \(1-i\). It tells us two things: the main result of the division is 1, and the expression didn't divide perfectly, hence a remainder is present.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition or multiplication. They might look simple like \(x+1\) or could be more complicated.Here are the main elements:

• Variables: Symbols like \(x\) or \(y\) that stand in for unknown numbers.
• Constants: Fixed numbers like 3 or -7.
• Operations: Can be addition, subtraction, multiplication, or division.

In expressions, especially when manipulating them, you follow hierarchies or rules, such as order of operations, to ensure correct answers. By using synthetic division on an expression like \(x+1\), even with a complex divisor like \(x+i\), you dig into these fundamental concepts, making the connections between variables and constants more tangible.