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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #2021 : Mathematical Relationships And Basic Graphs

Simplify the expression.

4i^2-6i-7i^2+3i+4

Possible Answers:

7-3i

-3i^2-3i+4

None of the other answer choices are correct.

-7-3i

Correct answer:

7-3i

Explanation:

Combine like terms. Treat $\small i$ as if it were any other variable.

4i^2-6i-7i^2+3i+4

-3i^2-3i+4

Substitute to eliminate $\small i^2$ .

i^2=-1

-3(-1)-3i+4

Simplify.

3-3i+4=7-3i

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Example Question #1 : Basic Operations With Complex Numbers

What is the value of (3 + i)(4 + i) ?

Possible Answers:

11 + 7i

12 + 8i

12 + 6i

Correct answer:

11 + 7i

Explanation:

When dealing with imaginary numbers, we multiply by foiling as we do with binomials. When we do this we get the expression below:

(3+i)(4+i)= 12 + 3i + 4i + i^2

Since we know that i^2 = -1 we get 12 + 7i - 1 which gives us 11 + 7i .

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Example Question #1 : Basic Operations With Complex Numbers

What is the value of i^4 ?

Possible Answers:

i + 1

Correct answer:

Explanation:

Recall that the definition of imaginary numbers gives that $i = \sqrt{-1}$ and thus that i^2 = -1 . Therefore, we can use Exponent Rules to write $i^4 = i^2 \cdot i^2 = -1\cdot-1 = 1$

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Example Question #2022 : Mathematical Relationships And Basic Graphs

$\frac{1-7i}{6-2i}=a-i$

Find .

Possible Answers:

0.5

2.5

Correct answer:

0.5

Explanation:

Multiply the numerator and denominator by the numerator's complex conjugate.

$\frac{1-7i}{6-2i}\ast \frac{6+2i}{6+2i}=\frac{20-40i}{40}$

Reduce/simplify.

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Example Question #2023 : Mathematical Relationships And Basic Graphs

Subtract:

(-1+5i)-(2-3i)

Possible Answers:

-3+8i

-3+2i

-3-8i

-3-2i

3-8i

Correct answer:

-3+8i

Explanation:

This is essentially the following expression after translation:

(-1+5i)-(2-3i)=-1-2+5i+3i

Now add the real parts together for a sum of , and add the imaginary parts for a sum of .

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Example Question #2024 : Mathematical Relationships And Basic Graphs

Multiply:

(2+3i)(1-i)

Answer must be in standard form.

Possible Answers:

5-i

5+i

2-3i

2+2i

Correct answer:

5+i

Explanation:

(2+3i)(1-i)

The first step is to distribute which gives us:

$2-2i+3i-3i^{2}$

$2+i-3i^{2}=2+i+3=5+i$

which is in standard form.

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Example Question #1 : Complex Numbers

Add:

(2-3i)+(-1-2i)

Possible Answers:

5-3i

1-5i

5+3i

-1+5i

1+5i

Correct answer:

1-5i

Explanation:

When adding complex numbers, add the real parts and the imaginary parts separately to get another complex number in standard form.

Adding the real parts gives 2-1=1 , and adding the imaginary parts gives -5i .

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Example Question #101 : Classifying Algebraic Functions

Divide: $\frac{2-i}{3+2i}$

The answer must be in standard form.

Possible Answers:

$\frac{2}{3}-\frac{1}{2}i$

$\frac{4}{13}-\frac{7i}{13}$

$\frac{1}{5i}$

$\frac{2}{3}i$

Correct answer:

$\frac{4}{13}-\frac{7i}{13}$

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator which is 3-2i which results in

$\frac{\left ( 2-i \right )\left ( 3-2i \right )}{\left ( 3+2i \right )\left ( 3-2i \right )}$

The numerator after simplification give us $6-4i-3i+2i^{2}=6-7i+2i^{2}=4-7i$

The denominator is equal to $3^{2}-4i^{2}=9+4=13$

Hence, the final answer in standard form =

$\frac{4}{13}-\frac{7i}{13}$

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Example Question #2 : Complex Numbers

Divide: $\frac{2+3i}{i}$

Answer must be in standard form.

Possible Answers:

3+2i

3-2i

2i-3

Correct answer:

3-2i

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator which is resulting in

$\frac{\left ( 2+3i \right )\left ( -i \right )}{\left ( i \right )\left ( -i \right )}$

This is equal to $\frac{-2i-{3i}^2}{-i^2}$

Since i^2 = -1 you can make that substitution of in place of i^2 in both numerator and denominator, leaving:

$\frac{-2i-3(-1)}{-(-1)}$

When you then cancel the negatives in both numerator and denominator (remember that -(-1)=1 , simplifying each term), you're left with a denominator of and a numerator of -2i + 3 , which equals 3-2i .

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Example Question #2025 : Mathematical Relationships And Basic Graphs

Evaluate: i(1+i)(2-i)

Possible Answers:

-1+3i

1+3 i

-1-3i

1-3i

-3+i

Correct answer:

-1+3i

Explanation:

Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

i(1+i)(2-i)

=i[(1)(2)+(1)(-i)+(i)(2)+(i)(-i)]

=i[2-i+2i-i^2]

=i[2+i-i^2]

=2i+i^2-i^3

The imaginary is equal to:

$\sqrt{-1}$

Write the terms for $i, i^2, \textup{and } i^3$ .

$i=\sqrt{-1}$

i^2=-1

$i^3=i \times i^2=-\sqrt{-1}=-i$

Replace $i^2 \textup{ and } i^3$ with the appropiate values and simplify.

2i+i^2-i^3= 2i-1-(-i) = -1+3i

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #2021 : Mathematical Relationships And Basic Graphs

Example Question #1 : Basic Operations With Complex Numbers

Example Question #1 : Basic Operations With Complex Numbers

Example Question #2022 : Mathematical Relationships And Basic Graphs

Example Question #2023 : Mathematical Relationships And Basic Graphs

Example Question #2024 : Mathematical Relationships And Basic Graphs

Example Question #1 : Complex Numbers

Example Question #101 : Classifying Algebraic Functions

Example Question #2 : Complex Numbers

Example Question #2025 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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