Problem 75 Find each of the products and ex... [FREE SOLUTION]

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Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 6: Problem 75

Find each of the products and express the answers in the standard form of a complex number. $$ (9+6 i)(-1-i) $$

Short Answer

Expert verified

The product is $-3 - 15i$.

Step by step solution

Apply the Distributive Property

First, apply the distributive property (also known as the FOIL method for two binomials) to expand the product:\[(9 + 6i)(-1 - i) = 9(-1) + 9(-i) + 6i(-1) + 6i(-i)\]

Perform Multiplication

Next, perform each multiplication:\[9(-1) = -9\]\[9(-i) = -9i\]\[6i(-1) = -6i\]\[6i(-i) = -6i^2\]

Simplify Using i虏 = -1

Remember that $i^2 = -1$, so simplify $-6i^2$ to get 6:\[-6i^2 = -6(-1) = 6\]

Combine Like Terms

Now combine all terms from the previous steps:\[ -9 - 9i - 6i + 6\]Combine the real parts and the imaginary parts separately:Real part: $-9 + 6 = -3$Imaginary part: $-9i - 6i = -15i$

Final Answer in Standard Form

Combine the simplified real part and imaginary part to write the complex number in standard form:\[-3 - 15i\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by terms inside a parenthesis. When we're dealing with complex numbers, this property will extend to multiplying pairs of binomials. In our exercise, we have two binomials:

The first is $9 + 6i$ where $9$ is the real part, and $6i$ is the imaginary part.
The second is $-1 - i$, where $-1$ is the real part, and $-i$ is the imaginary part.

To multiply these binomials, we apply the distributive property, effectively multiplying each term from the first binomial by each term in the second binomial. We get:
$9(-1) + 9(-i) + 6i(-1) + 6i(-i)$.
By distributing this will yield four separate terms that we can then simplify and combine.

FOIL Method

The FOIL method is a specific application of the distributive property that helps with multiplying two binomials. The acronym "FOIL" stands for:

First: Multiply the first terms in each binomial;
Outer: Multiply the outer terms;
Inner: Multiply the inner terms;
Last: Multiply the last terms.

Applying the FOIL method to $(9 + 6i)(-1 - i)$, we calculate:
- First: - $9 \times (-1) = -9$
- Outer: - $9 \times (-i) = -9i$
- Inner: - $6i \times (-1) = -6i$
- Last: - $6i \times (-i) = -6i^2$
Each part is calculated separately, and then we combine the results to simplify the expression.

Imaginary Unit i

The imaginary unit $i$ is a crucial concept in complex numbers. It stands for the square root of -1. Knowing the property $i^2 = -1$ is essential when dealing with complex numbers. In the multiplication problem $6i \times (-i)$, we have:

The direct result is $-6i^2$; because $i^2 = -1$, we can substitute $-1$ for $i^2$.
Simplifying gives:$-6(-1) = 6$,changing the sign to the opposite.

This simplification turns the expression into something combineable with other real parts of the expression. Remembering that $i^2 = -1$ can help eliminate $i^2$ from your calculation and clarify the terms you need to combine.

Binomials Expansion

Expanding binomials involves taking two binomials and multiplying them to form a new polynomial. When dealing with complex numbers, this includes both real and imaginary parts. For$(9 + 6i)(-1 - i)$, we expand each binomial term-by-term, leading to the expression:\[-9 - 9i - 6i + 6\].To simplify, it's important to:

Combine like terms 鈥� those that are similar in such nature, like real numbers with real numbers and imaginary with imaginary.
Real parts: - Combine $-9 + 6 = -3$
Imaginary parts: - Combine $-9i - 6i = -15i$

Finally, you write the resulting expression in standard complex number notation, which is $-3 - 15i$. By fully expanding and simplifying the binomials, we clearly see both the process and the solution, essential in algebra for understanding operations with complex numbers.

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91影视

Find each of the products and express the answers in the standard form of a complex number. $$ (9+6 i)(-1-i) $$

Short Answer

Step by step solution

Apply the Distributive Property

Perform Multiplication

Simplify Using i虏 = -1

Combine Like Terms

Final Answer in Standard Form

Key Concepts

Distributive Property

FOIL Method

Imaginary Unit i

Binomials Expansion

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