Select The Correct Answer From Each Drop-down Menu.What Is The Factored Form Of This Expression? 27 T 3 − 36 T 2 − 12 T + 16 = ( □ ∨ ) ( □ ∨ ) ( □ ∨ 27t^3 - 36t^2 - 12t + 16 = (\square \vee)(\square \vee)(\square \vee 27 T 3 − 36 T 2 − 12 T + 16 = ( □ ∨ ) ( □ ∨ ) ( □ ∨ ]

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Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will guide you through the process of factoring the given expression: 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16. Our goal is to rewrite this expression in its factored form, which will be represented as (□∨)(□∨)(□∨)(\square \vee)(\square \vee)(\square \vee).

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring an expression is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides each term without leaving a remainder. In this case, we can see that the GCF of the given expression is 1, since there is no common factor that divides all the terms.

Step 2: Look for Common Factors

Since the GCF is 1, we need to look for other common factors. We can start by factoring out the greatest common factor of the first two terms, which is 9t29t^2. We can rewrite the expression as:

27t3−36t2−12t+16=9t2(3t−4)−12t+1627t^3 - 36t^2 - 12t + 16 = 9t^2(3t - 4) - 12t + 16

Step 3: Factor Out the Common Binomial Factor

Now, we can see that the expression can be factored further by taking out the common binomial factor (3t−4)(3t - 4). We can rewrite the expression as:

27t3−36t2−12t+16=9t2(3t−4)−4(3t−4)27t^3 - 36t^2 - 12t + 16 = 9t^2(3t - 4) - 4(3t - 4)

Step 4: Factor Out the Common Binomial Factor Again

We can see that the expression can be factored further by taking out the common binomial factor (3t−4)(3t - 4). We can rewrite the expression as:

27t3−36t2−12t+16=(3t−4)(3t2−4t−4)27t^3 - 36t^2 - 12t + 16 = (3t - 4)(3t^2 - 4t - 4)

Step 5: Factor the Quadratic Expression

The quadratic expression 3t2−4t−43t^2 - 4t - 4 can be factored further by finding two numbers whose product is −12-12 and whose sum is −4-4. These numbers are −6-6 and 22. We can rewrite the quadratic expression as:

3t2−4t−4=(3t+2)(t−2)3t^2 - 4t - 4 = (3t + 2)(t - 2)

Step 6: Write the Factored Form of the Expression

Now that we have factored the quadratic expression, we can write the factored form of the original expression as:

27t3−36t2−12t+16=(3t−4)(3t+2)(t−2)27t^3 - 36t^2 - 12t + 16 = (3t - 4)(3t + 2)(t - 2)

Conclusion

In this article, we have guided you through the process of factoring the given expression: 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16. We have identified the greatest common factor (GCF) of the expression, looked for common factors, factored out the common binomial factor, and finally factored the quadratic expression. The factored form of the expression is (3t−4)(3t+2)(t−2)(3t - 4)(3t + 2)(t - 2).

Final Answer

The final answer is: (3t−4)(3t+2)(t−2)\boxed{(3t - 4)(3t + 2)(t - 2)}

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Introduction

In our previous article, we guided you through the process of factoring the expression: 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16. We identified the greatest common factor (GCF) of the expression, looked for common factors, factored out the common binomial factor, and finally factored the quadratic expression. In this article, we will answer some frequently asked questions (FAQs) related to factoring expressions.

Q&A

Q: What is the greatest common factor (GCF) of an expression?

A: The greatest common factor (GCF) of an expression is the largest factor that divides each term without leaving a remainder.

Q: How do I identify the GCF of an expression?

A: To identify the GCF of an expression, look for the largest factor that divides each term without leaving a remainder. You can also use the following steps:

  1. List all the factors of each term.
  2. Identify the common factors among the terms.
  3. Choose the largest common factor as the GCF.

Q: What is a common factor?

A: A common factor is a factor that divides each term of an expression without leaving a remainder.

Q: How do I factor out a common binomial factor?

A: To factor out a common binomial factor, follow these steps:

  1. Identify the common binomial factor among the terms.
  2. Write the expression as a product of the common binomial factor and the remaining terms.
  3. Simplify the expression.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, follow these steps:

  1. Identify the coefficients of the quadratic expression.
  2. Find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
  3. Write the quadratic expression as a product of two binomials.

Q: What is the difference between factoring and simplifying an expression?

A: Factoring an expression involves expressing it as a product of simpler expressions, while simplifying an expression involves combining like terms to reduce its complexity.

Q: Why is factoring important in algebra?

A: Factoring is important in algebra because it allows us to simplify complex expressions, identify common factors, and solve equations.

Q: Can I factor an expression with a negative sign?

A: Yes, you can factor an expression with a negative sign. When factoring an expression with a negative sign, remember to include the negative sign in the factored form.

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

A: To determine if an expression can be factored, look for common factors, binomial factors, or quadratic expressions. If you can identify any of these, you can factor the expression.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to factoring expressions. We have covered topics such as the greatest common factor (GCF), common factors, factoring out binomial factors, quadratic expressions, and the importance of factoring in algebra. We hope that this article has provided you with a better understanding of factoring expressions and has helped you to improve your algebra skills.

Final Answer

The final answer is: \boxed{There is no final numerical answer to this article. The article provides a Q&A guide to factoring expressions.}