Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the algebraic expressions. If \(y=\frac{1+2 x}{x+3},\) evaluate \(y\) given \(x=4 i\).

Step-by-Step Solution

Answer

\( y = \frac{7}{5} + \frac{4i}{5} \) for \( x = 4i \).

Step 1: Understand the expression

The given algebraic expression is \( y = \frac{1+2x}{x+3} \). You need to substitute \( x = 4i \) into the expression and evaluate \( y \).

Step 2: Substitute the given value for x

Replace the variable \( x \) in the expression with the value \( 4i \): \[ y = \frac{1 + 2(4i)}{4i + 3} \]

Step 3: Simplify the numerator

Calculate \( 2(4i) \) which is \( 8i \), then simplify the numerator: \( 1 + 8i \). So now the expression is: \( y = \frac{1 + 8i}{4i + 3} \)

Step 4: Rationalize the denominator

To simplify the expression, multiply the numerator and the denominator by the conjugate of the denominator, \( 3 - 4i \): \[ y = \frac{(1 + 8i)(3 - 4i)}{(4i + 3)(3 - 4i)} \]

Step 5: Multiply the terms

First, calculate the denominator: \((4i + 3)(3 - 4i) = 9 + 12i - 12i - 16i^2 = 25\).Now, calculate the numerator: \((1 + 8i)(3 - 4i) = 3 - 4i + 24i - 32i^2 = 3 + 20i + 32 = 35 + 20i\).

Step 6: Combine and simplify the real and imaginary components

Given that \(i^2 = -1\), the expression simplifies to: \( y = \frac{35 + 20i}{25} \), now divide each term by 25: \[ y = \frac{35}{25} + \frac{20i}{25} = \frac{7}{5} + \frac{4i}{5} \]

Step 7: Final Answer: Simplified Result

The expression \( y \) evaluates to \( \frac{7}{5} + \frac{4i}{5} \) when \( x = 4i \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations such as addition, subtraction, multiplication, and division. In this exercise, the expression is given as \( y = \frac{1+2x}{x+3} \). This expression combines constants, the variable \( x \), and operations.When working with algebraic expressions, your primary tasks might include:

• Substitution: Replacing variables with given values.
• Simplifying: Combining like terms and arranging expressions in a simpler form.

In the context of this problem, we substituted the complex number \( x = 4i \) into the expression and simplified it step by step.

Rationalization

Rationalization involves eliminating irrational numbers (like square roots or complex expressions) from the denominator of a fraction. In this problem, we are tasked with rationalizing the denominator of a complex fraction where \( x = 4i \).The denominator \( 4i + 3 \) is rationalized by multiplying both the numerator and the denominator by the conjugate of the denominator, \( 3 - 4i \). The conjugate flips the sign of the imaginary part:

• Denominator: \((4i + 3)(3 - 4i) = 25\)
• Numerator: \((1 + 8i)(3 - 4i) = 35 + 20i\)

Through this process, the denominator becomes real, allowing us to simplify the fraction further.

Imaginary Unit

The imaginary unit \( i \) is defined as \( i^2 = -1 \), which forms the basis for working with complex numbers. In this exercise, \( x = 4i \), which introduces an imaginary component into our algebraic expression.Using \( i \) allows us to work in the complex plane, where each number has a real and an imaginary component. Here’s how it affects our calculation:

• When multiplying terms involving \( i \), use the property \( i^2 = -1 \).
• In the denominator, \( (4i)(-4i) = 16i^2 = -16 \), significantly simplifying the expression.

Understanding \( i \) is crucial for simplifying expressions correctly involving complex numbers.

Polynomial Division

Although this problem primarily involves complex numbers and rationalization, understanding polynomial division can help simplify expressions involving fractions. Polynomial division is the process of dividing a polynomial by another polynomial. Here, we simplify to achieve a form with a real number in the denominator. Steps include:

• Rationalizing to clear complex denominators.
• Simplifying terms by performing polynomial-like division (though this specific division isn't present in the exercise, understanding the concept can aid similar tasks).

It's about breaking down complex expressions into simpler components we can more easily understand and work with.