Factor Completely $x^2 + 16$.A. \[$(x+4)(x+4)\$\]B. \[$(x+4)(x-4)\$\]C. PrimeD. \[$(x-4)(x-4)\$\]

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we will focus on factoring the quadratic expression $x^2 + 16$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor this expression.

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two binomial expressions. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. For example, $x^2 + 4x + 4$ is a quadratic expression. Factoring involves finding two binomial expressions that, when multiplied together, result in the original quadratic expression.

Methods of Factoring

There are several methods of factoring quadratic expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the quadratic expression into two groups and then factoring out the greatest common factor (GCF) from each group.
• Factoring by Difference of Squares: This method involves recognizing that the quadratic expression can be written as a difference of squares, which can then be factored into two binomial expressions.
• Factoring by Perfect Square Trinomials: This method involves recognizing that the quadratic expression can be written as a perfect square trinomial, which can then be factored into two binomial expressions.

Factoring $x^2 + 16$

To factor the quadratic expression $x^2 + 16$, we can use the method of factoring by difference of squares. This method involves recognizing that the quadratic expression can be written as a difference of squares, which can then be factored into two binomial expressions.

Step 1: Identify the Difference of Squares

The quadratic expression $x^2 + 16$ can be written as a difference of squares:

$$x^2 + 16 = (x)^2 - (4i)^2$$

Step 2: Factor the Difference of Squares

The difference of squares can be factored into two binomial expressions:

$$(x)^2 - (4i)^2 = (x + 4i)(x - 4i)$$

Step 3: Simplify the Expression

The expression $(x + 4i)(x - 4i)$ can be simplified by multiplying the two binomial expressions:

$$(x + 4i)(x - 4i) = x^2 - (4i)^2$$

Conclusion

In conclusion, the quadratic expression $x^2 + 16$ can be factored using the method of factoring by difference of squares. The factored form of the expression is $(x + 4i)(x - 4i)$.

Answer

The correct answer is:

• A. {(x+4)(x+4)$}$ is incorrect because it does not account for the imaginary unit $i$.
• B. {(x+4)(x-4)$}$ is incorrect because it does not account for the imaginary unit $i$.
• C. Prime is incorrect because the expression can be factored further.
• D. {(x-4)(x-4)$}$ is incorrect because it does not account for the imaginary unit $i$.

$$$$$$$$

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we will provide a Q&A guide on factoring quadratic expressions, including common mistakes and tips for success.

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two binomial expressions. A quadratic expression is a polynomial of degree two, which means it has a highest power of two.

Q: What are the different methods of factoring?

A: There are several methods of factoring quadratic expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the quadratic expression into two groups and then factoring out the greatest common factor (GCF) from each group.
• Factoring by Difference of Squares: This method involves recognizing that the quadratic expression can be written as a difference of squares, which can then be factored into two binomial expressions.
• Factoring by Perfect Square Trinomials: This method involves recognizing that the quadratic expression can be written as a perfect square trinomial, which can then be factored into two binomial expressions.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use one of the methods mentioned above. Here are the steps to follow:

1. Identify the type of quadratic expression: Determine if the quadratic expression can be factored using one of the methods mentioned above.
2. Apply the method: Use the method to factor the quadratic expression.
3. Simplify the expression: Simplify the factored expression to get the final answer.

Q: What are some common mistakes to avoid when factoring?

A: Here are some common mistakes to avoid when factoring:

• Not identifying the type of quadratic expression: Make sure to identify the type of quadratic expression before attempting to factor it.
• Not applying the correct method: Use the correct method to factor the quadratic expression.
• Not simplifying the expression: Simplify the factored expression to get the final answer.

Q: What are some tips for success when factoring?

A: Here are some tips for success when factoring:

• Practice, practice, practice: Practice factoring quadratic expressions to become more comfortable with the process.
• Use the correct method: Use the correct method to factor the quadratic expression.
• Simplify the expression: Simplify the factored expression to get the final answer.

Q: Can you provide an example of factoring a quadratic expression?

A: Here is an example of factoring a quadratic expression:

Example:

Factor the quadratic expression $x^2 + 16$.

Solution:

To factor the quadratic expression $x^2 + 16$, we can use the method of factoring by difference of squares. This method involves recognizing that the quadratic expression can be written as a difference of squares, which can then be factored into two binomial expressions.

Step 1: Identify the difference of squares

The quadratic expression $x^2 + 16$ can be written as a difference of squares:

$$x^2 + 16 = (x)^2 - (4i)^2$$

Step 2: Factor the difference of squares

The difference of squares can be factored into two binomial expressions:

$$(x)^2 - (4i)^2 = (x + 4i)(x - 4i)$$

Step 3: Simplify the expression

The expression $(x + 4i)(x - 4i)$ can be simplified by multiplying the two binomial expressions:

$$(x + 4i)(x - 4i) = x^2 - (4i)^2$$

Conclusion:

In conclusion, the quadratic expression $x^2 + 16$ can be factored using the method of factoring by difference of squares. The factored form of the expression is $(x + 4i)(x - 4i)$.

Answer:

The correct answer is:

• A. {(x+4)(x+4)$}$ is incorrect because it does not account for the imaginary unit $i$.
• B. {(x+4)(x-4)$}$ is incorrect because it does not account for the imaginary unit $i$.
• C. Prime is incorrect because the expression can be factored further.
• D. {(x-4)(x-4)$}$ is incorrect because it does not account for the imaginary unit $i$.

The correct answer is A. {(x+4i)(x-4i)$}$, but since the options do not include the imaginary unit $i$, the closest correct answer is A. {(x+4)(x+4)$}$, but this is incorrect because it does not account for the imaginary unit $i$.