Select The Correct Answer From Each Drop-down Menu.What Is The Factored Form Of This Expression? ${ 27t^3 - 36t^2 - 12t + 16 = (\square)(\square)(\square) }$

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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 focus on factoring the given expression 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16 and finding its factored form.

Understanding the Expression

Before we begin factoring, let's take a closer look at the given expression. We have a cubic polynomial with four terms: 27t327t^3, −36t2-36t^2, −12t-12t, and 1616. Our goal is to express this polynomial as a product of simpler polynomials.

Factoring by Grouping

One of the techniques used to factor a polynomial is factoring by grouping. This method involves grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair.

Step 1: Group the Terms

Let's group the terms of the polynomial into pairs:

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

Step 2: Factor Out the GCF

Now, let's factor out the GCF from each pair:

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

Step 3: Factor Out the Common Factor

We can see that both pairs have a common factor of (3t−4)(3t - 4). Let's factor it out:

9t2(3t−4)−4(3t−4)=(3t−4)(9t2−4)9t^2(3t - 4) - 4(3t - 4) = (3t - 4)(9t^2 - 4)

Factoring the Quadratic

Now, we have a quadratic expression 9t2−49t^2 - 4 that we need to factor. We can use the difference of squares formula to factor it:

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

In this case, a=3ta = 3t and b=2b = 2. So, we can write:

9t2−4=(3t)2−22=(3t+2)(3t−2)9t^2 - 4 = (3t)^2 - 2^2 = (3t + 2)(3t - 2)

Factored Form

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

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

Conclusion

In this article, we have factored the given expression 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16 and found its factored form. We used the technique of factoring by grouping and the difference of squares formula to factor the quadratic expression. The factored form of the polynomial is (3t−4)(3t+2)(3t−2)(3t - 4)(3t + 2)(3t - 2).

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Introduction

In our previous article, we factored the expression 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16 and found its factored form. In this article, we will answer some common questions related to factoring expressions.

Q&A

Q: What is factoring an expression?

A: Factoring an expression is a process of expressing a polynomial as a product of simpler polynomials.

Q: Why is factoring important?

A: Factoring is an important concept in algebra because it helps us to simplify complex expressions and solve equations.

Q: What are the different methods of factoring?

A: There are several methods of factoring, including factoring by grouping, factoring out the greatest common factor (GCF), and using the difference of squares formula.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use the difference of squares formula: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b).

Q: What is the difference of squares formula?

A: The difference of squares formula is a formula that allows us to factor a quadratic expression in the form of a2−b2a^2 - b^2.

Q: How do I factor a polynomial with multiple terms?

A: To factor a polynomial with multiple terms, you can use the technique of factoring by grouping. This involves grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair.

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

A: The greatest common factor (GCF) is the largest factor that divides each term of a polynomial.

Q: How do I find the GCF of a polynomial?

A: To find the GCF of a polynomial, you can list the factors of each term and then find the greatest common factor.

Q: What is the factored form of the expression 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16?

A: The factored form of the expression 27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16 is (3t−4)(3t+2)(3t−2)(3t - 4)(3t + 2)(3t - 2).

Conclusion

In this article, we have answered some common questions related to factoring expressions. We have discussed the different methods of factoring, including factoring by grouping and using the difference of squares formula. We have also provided examples of how to factor a quadratic expression and a polynomial with multiple terms.