Problem 7 - College Algebra

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

Step 1: Substitute the Given Value for x

Start by substituting the given complex number \(x = 2 + i\) into the expression for \(y = x^2 + 3x + 5\). This gives us: \(y = (2+i)^2 + 3(2+i) + 5\).

Step 2: Expand the Square

Next, expand \((2+i)^2\). Applying the formula \((a+b)^2 = a^2 + 2ab + b^2\), we get: \((2+i)^2 = 2^2 + 2(2)(i) + i^2 = 4 + 4i + i^2\). Since \(i^2 = -1\), this simplifies to \(4 + 4i - 1 = 3 + 4i\).

Step 3: Multiply by 3

Now, calculate \(3(2+i)\). Distribute the 3: \(3(2+i) = 3 \times 2 + 3 \times i = 6 + 3i\).

Step 4: Substitute and Simplify

Substitute the results from the previous steps back into the expression: \(y = (3 + 4i) + (6 + 3i) + 5\). Simplify by adding like terms: Real parts: \(3 + 6 + 5 = 14\). Imaginary parts: \(4i + 3i = 7i\). Thus, \(y = 14 + 7i\).

Key Concepts

Evaluating Algebraic Expressions

Evaluating algebraic expressions involves substituting given values into the expression and simplifing the result. In the exercise provided, we are substituting a complex number into a polynomial expression. Here's a step-by-step guide to understanding and evaluating such expressions:

• **Identify the expression:** Know what needs to be evaluated. In this case, it's the equation for \(y\): \(y = x^2 + 3x + 5\).
• **Substitute the given value:** Plug the specific value given for \(x\), which is the complex number \(2+i\), into the expression.
• **Simplify the expression:** After substituting the value, carry out arithmetic operations—addition, subtraction, multiplication—in the expression, carefully keeping track of both real and imaginary parts.

When working with complex numbers, it's essential to handle the imaginary part, denoted by \(i\), appropriately. For example, knowing that \(i^2 = -1\) allows you to replace \(i^2\) with \(-1\) whenever it appears in your calculations.

Complex Arithmetic

Complex arithmetic involves performing mathematical operations—such as addition, subtraction, multiplication, and division—with complex numbers. These operations are similar to arithmetic with real numbers, but with some unique properties due to the presence of an imaginary unit \(i\).

• **Addition and Subtraction:** Treat real and imaginary parts separately. If you have two complex numbers \(a+bi\) and \(c+di\), they add up to \((a+c) + (b+d)i\).
• **Multiplication:** Use the distributive property, similar to multiplying binomials. For example, \((a+bi)(c+di) = ac + adi + bci + bidi\). Remember that \(i^2 = -1\), which helps simplify terms.
• **Dividing Complex Numbers:** Often involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

In the problem we solved, the multiplication of \(3(2+i)\) and expansion of \((2+i)^2\) both required handling the imaginary unit \(i\). Understanding these basic operations is crucial for working with complex expressions.

Polynomial Expressions

Polynomial expressions consist of terms made up of variables raised to whole-number exponents, combined using addition, subtraction, and multiplication. Evaluating polynomials involves substituting values into the variables and simplifying.

• **Structure of Polynomial:** A polynomial has terms such as \(x^2, x, \) and constant terms like 5 in this exercise.
• **Substitution:** Insert the given value into each term. For instance, \(x=2+i\) is substituted into each occurrence of \(x\) in \(x^2 + 3x + 5\).
• **Combining Like Terms:** Once values are substituted, combine the like terms—the constant and the coefficients of the variable terms.

Polynomials can be evaluated using complex numbers just as they are with real numbers. Each component of the polynomial, when expanded and simplified correctly, deals with the complex arithmetic to provide the final result, as we found \(y = 14 + 7i\). Comprehending this process allows for tackling more profound and greater complexity in algebraic and polynomial problems.