Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the algebraic expressions. If \(y=x^{2}+3 x+5,\) evaluate \(y\) given \(x=2+i\).

Step-by-Step Solution

Answer

The evaluated expression is \( y = 14 + 7i \).

Step 1: Substitute the value of x

Start with the given expression for \( y \): \[ y = x^2 + 3x + 5 \] Substitute \( x = 2+i \) into the expression. The expression becomes: \[ y = (2+i)^2 + 3(2+i) + 5 \]

Step 2: Expand the square term

Calculate \((2+i)^2\):\[(2+i)^2 = (2+i)(2+i) = 4 + 2i + 2i + i^2 = 4 + 4i + i^2 \]Since \(i^2 = -1\), replace \(i^2\) to get:\[4 + 4i - 1 = 3 + 4i\]

Step 3: Expand the coefficient of 3

Distribute 3 over \(2+i\):\[3(2+i) = 6 + 3i\]

Step 4: Combine all parts together

Now, add together the results of each expanded term with the constant 5:\[y = (3 + 4i) + (6 + 3i) + 5\]First, combine the real parts: \[3 + 6 + 5 = 14\]Then, combine the imaginary parts:\[4i + 3i = 7i\]

Step 5: Write the final answer

Combine the results from Step 4: \[ y = 14 + 7i \]This is the final result for the given expression.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The real part is \(a\), and the imaginary part is \(bi\).

• In the context of the exercise, \(2 + i\) is a complex number.
• The real part is 2, and the imaginary part is \(i\), which represents \(1i\).

Understanding complex numbers is crucial as they expand our ability to solve equations that have no real solutions. When you encounter terms like \((2+i)^2\), it involves both the real and imaginary components simultaneously.

Polynomial Expressions

Polynomial expressions are expressions that involve variables raised to various powers, combined using addition, subtraction, and multiplication. In our exercise, the expression \(y = x^2 + 3x + 5\) is a polynomial of degree 2 with coefficients 1, 3, and 5. The degree is determined by the highest power of the variable in the expression.

• Polynomials are commonly used in algebra to represent complex relationships.
• They can have real coefficients or even complex numbers as coefficients.

To evaluate polynomial expressions, you substitute specific values for the variables and perform calculations to simplify. This process applies the properties of numbers and the operations in the expression.

Substitution Method

The substitution method involves replacing variables in an expression with given values to simplify and solve it. This method is key in evaluating polynomial expressions and is very handy when working with specific scenarios or solving equations.

• In this exercise, we substitute \(x = 2 + i\) into the polynomial \(y = x^2 + 3x + 5\).
• This substitution turns our polynomial into an expression involving complex numbers.

After substitution, the expression can be simplified using arithmetic of complex numbers to find the value of \(y\), which will also be a complex number as shown in the solution steps.

Imaginary Unit

The imaginary unit \(i\) is foundational to complex numbers. It is defined as the square root of \(-1\), such that \(i^2 = -1\). This property is essential when calculating powers of complex numbers and simplifying expressions that involve them.

• When you see expressions like \((2+i)^2\), you need to use the fact that \(i^2 = -1\) in your calculations.
• It helps convert imaginary terms into real numbers or other simpler forms.

Recognizing this definition allows you to correctly combine and simplify terms in expressions involving \(i\), which is crucial for carrying out the simplification process correctly.

Step-by-Step Solution

A step-by-step solution is an essential method to solve complex problems, such as evaluating algebraic expressions. It involves breaking down each part of the question into manageable steps, carefully performing calculations at each stage. This approach enhances understanding and ensures accuracy.

• It starts with substituting given values for variables, as seen in the exercise where \(2+i\) was substituted for \(x\).
• Each subsequent step takes you closer to a complete solution by progressively simplifying the expression.
• Finally, combining terms and applying arithmetic rules results in the evaluated expression: \(y = 14 + 7i\).

By meticulously going through each step, one can thoroughly understand the process, making it easier to tackle similar problems in the future.