Problem 71 - Algebra 2

Question

Simplify each expression. \((4-3 i)(5+i)\)

Step-by-Step Solution

Answer

The simplified form of the expression is \(23 - 11i\).

Step 1: Apply the Distributive Property

Use the distributive property to multiply each term in the first complex number by each term in the second complex number. Perform the operations: \((4 \times 5) + (4 \times i) + (-3i \times 5) + (-3i \times i)\).

Step 2: Multiply the Real and Imaginary Parts

Carry out the multiplication for each pair of terms: \(20 + 4i - 15i - 3i^2\). Remember that \(i^2 = -1\).

Step 3: Combine Like Terms and Simplify

Combine the real parts and the imaginary parts separately, and replace \(i^2\) with \(-1\): \(20 - 3(-1) + (4i - 15i)\). Then simplify the expression to combine like terms.

Step 4: Express the Simplified Form

After combining like terms and simplifying, we get the final simplified form of the given expression.

Key Concepts

Distributive Property

When simplifying complex numbers, the distributive property plays a crucial role. It allows us to multiply a single term by each term in a binomial. For instance, given a complex number multiplication like \((4-3i)(5+i)\), the distributive property dictates that we should multiply each part of the first complex number by each part of the second one.

Think of it like sharing slices of a pie evenly. Each slice (term) from the first complex number gets shared with every slice from the second. Mathematically, this gives us \((4 \times 5) + (4 \times i) + (-3i \times 5) + (-3i \times i)\). Each product represents a piece of the final answer you're putting together, like assembling a puzzle with both real numbers and imaginary units.

Real and Imaginary Parts

Complex numbers have two distinct parts: the real part and the imaginary part. In the complex number \(a + bi\), \(a\) is the real part, and \(bi\) is the imaginary part, where \(i\) represents the square root of \(-1\).

Understanding this is crucial when simplifying complex expressions. After applying the distributive property, as you perform the multiplication, ensure you accurately multiply the real and imaginary parts. For our example, this step provides you with real components such as \(20\) and \(-3i^2\), and imaginary parts like \(4i\) and \(-15i\). Remember the special rule that \(i^2 = -1\), which helps in converting terms like \(-3i^2\) into a real number to further simplify the expression.

Combining Like Terms

The final step in simplifying complex numbers is combining like terms. After distributing and multiplying, gather all your real parts on one side and your imaginary parts on another. From our earlier multiplication, you might see terms like \(20\) and \(-3\) (from \(-3i^2\), since \(i^2\) is -1) bear a resemblance – they are real numbers, and thus, like terms.

Similarly, combine \(4i\) and \(-15i\) since they’re both imaginary. Combining like terms helps you clean up the equation, making it more readable and easier to understand. In this case, it simplifies to \(20 - 3(-1) + (4i - 15i)\), which further simplifies to the final form once you add and subtract these figures, keeping the real and imaginary components intact.