Factor Completely:A. X 4 − 13 X 2 + 36 X^4 - 13x^2 + 36 X 4 − 13 X 2 + 36 B. 4 X 4 − 13 X 2 + 9 4x^4 - 13x^2 + 9 4 X 4 − 13 X 2 + 9

Introduction

Factoring is a fundamental concept in algebra that allows us to simplify complex expressions by breaking them down into their most basic components. In this article, we will explore how to factor completely two given expressions: $x^4 - 13x^2 + 36$ and $4x^4 - 13x^2 + 9$. We will delve into the techniques and strategies used to factor these expressions, and provide step-by-step solutions to help you understand the process.

Factoring Expression A: $x^4 - 13x^2 + 36$

Step 1: Identify the Type of Expression

The given expression $x^4 - 13x^2 + 36$ is a quadratic expression in terms of $x^2$. This means we can treat $x^2$ as a single variable and apply the techniques used to factor quadratic expressions.

Step 2: Factor the Expression

To factor the expression, we need to find two numbers whose product is $36$ and whose sum is $-13$. These numbers are $-9$ and $-4$, since $(-9) \times (-4) = 36$ and $(-9) + (-4) = -13$.

x^4 - 13x^2 + 36 = (x^2 - 9)(x^2 - 4)

Step 3: Factor the Quadratic Expressions

Now that we have factored the expression into two quadratic expressions, we can further factor them using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$.

(x^2 - 9) = (x - 3)(x + 3)
(x^2 - 4) = (x - 2)(x + 2)

Step 4: Write the Final Factored Form

Combining the results from the previous steps, we can write the final factored form of the expression:

x^4 - 13x^2 + 36 = (x - 3)(x + 3)(x - 2)(x + 2)

Factoring Expression B: $4x^4 - 13x^2 + 9$

Step 1: Identify the Type of Expression

The given expression $4x^4 - 13x^2 + 9$ is also a quadratic expression in terms of $x^2$. We can treat $x^2$ as a single variable and apply the techniques used to factor quadratic expressions.

Step 2: Factor the Expression

To factor the expression, we need to find two numbers whose product is $36$ and whose sum is $-13$. However, in this case, we are looking for numbers whose product is $36$ and whose sum is $-13$, but we also need to consider the coefficient of the $x^4$ term, which is $4$. This means we need to find two numbers whose product is $36$ and whose sum is $-13$, and then multiply the entire expression by $4$.

4x^4 - 13x^2 + 9 = 4(x^4 - 13x^2/4 + 9/4)

Step 3: Simplify the Expression

Simplifying the expression, we get:

4x^4 - 13x^2 + 9 = 4(x^4 - 13x^2/4 + 9/4)
= 4(x^2 - 9/4)(x^2 - 4)

Step 4: Factor the Quadratic Expressions

Now that we have factored the expression into two quadratic expressions, we can further factor them using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$.

(x^2 - 9/4) = (x - 3/2)(x + 3/2)
(x^2 - 4) = (x - 2)(x + 2)

Step 5: Write the Final Factored Form

Combining the results from the previous steps, we can write the final factored form of the expression:

4x^4 - 13x^2 + 9 = 4(x - 3/2)(x + 3/2)(x - 2)(x + 2)

Conclusion

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Introduction

In our previous article, we explored how to factor completely two given expressions: $x^4 - 13x^2 + 36$ and $4x^4 - 13x^2 + 9$. We delved into the techniques and strategies used to factor these expressions, and provided step-by-step solutions to help you understand the process. In this article, we will answer some of the most frequently asked questions related to factoring expressions.

Q&A

Q: What is factoring in algebra?

A: Factoring is a process of breaking down a complex expression into simpler components, called factors. These factors can be numbers, variables, or a combination of both.

Q: Why is factoring important in algebra?

A: Factoring is important in algebra because it allows us to simplify complex expressions, make them easier to work with, and solve equations more efficiently.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

• Greatest Common Factor (GCF) factoring: This involves finding the greatest common factor of two or more terms and factoring it out.
• Difference of Squares factoring: This involves factoring expressions of the form $a^2 - b^2$.
• Perfect Square Trinomial factoring: This involves factoring expressions of the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$.
• Quadratic Formula factoring: This involves factoring expressions of the form $ax^2 + bx + c$.

Q: How do I know which type of factoring to use?

A: To determine which type of factoring to use, you need to examine the expression and look for patterns. For example, if the expression is in the form $a^2 - b^2$, you can use the difference of squares formula.

Q: What is the difference between factoring and simplifying?

A: Factoring involves breaking down an expression into simpler components, while simplifying involves combining like terms to make an expression easier to work with.

Q: Can I factor an expression that has a variable in the denominator?

A: No, you cannot factor an expression that has a variable in the denominator. This is because the denominator must be a constant in order to factor the expression.

Q: How do I factor an expression with a coefficient?

A: To factor an expression with a coefficient, you need to multiply the entire expression by the reciprocal of the coefficient. For example, if the expression is $2x^2 + 5x + 3$, you can factor it by multiplying the entire expression by $1/2$.

Q: Can I factor an expression that has a negative sign in front of it?

A: Yes, you can factor an expression that has a negative sign in front of it. In fact, the negative sign can often be factored out along with the other terms.

Q: How do I know if an expression can be factored?

A: To determine if an expression can be factored, you need to examine the expression and look for patterns. If the expression can be broken down into simpler components, it can be factored.

Conclusion

In this article, we have answered some of the most frequently asked questions related to factoring expressions. We have covered topics such as the importance of factoring, the different types of factoring, and how to determine which type of factoring to use. By following these tips and techniques, you can become more confident in your ability to factor expressions and simplify complex algebraic expressions.