6.3 HW - Factoring Trinomials Factor Completely: $72m^2 - 17m - 72$.Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. $72m^2 - 17m - 72 =$ $\square$B. The Polynomial Is Prime.

Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will delve into the world of factoring trinomials, focusing on the specific problem of factoring the expression $72m^2 - 17m - 72$. We will explore the various techniques and strategies used to factor trinomials, and provide a step-by-step solution to the given problem.

What are Trinomials?

A trinomial is a polynomial expression with three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Trinomials can be factored using various techniques, including the greatest common factor (GCF) method, the grouping method, and the factoring by grouping method.

Factoring Trinomials: Techniques and Strategies

There are several techniques and strategies used to factor trinomials, including:

• Greatest Common Factor (GCF) Method: This method involves finding the greatest common factor of the three terms in the trinomial and factoring it out.
• Grouping Method: This method involves grouping the first two terms and the last two terms of the trinomial and factoring out the common factors.
• Factoring by Grouping Method: This method involves grouping the terms in the trinomial and factoring out the common factors.

Step-by-Step Solution to the Problem

To factor the expression $72m^2 - 17m - 72$, we can use the factoring by grouping method.

Step 1: Factor out the GCF

The first step is to factor out the greatest common factor (GCF) of the three terms in the trinomial. In this case, the GCF is 1.

Step 2: Group the Terms

Next, we group the first two terms and the last two terms of the trinomial.

$72m^2 - 17m - 72 = (72m^2 - 72) - 17m$

Step 3: Factor out the Common Factors

Now, we factor out the common factors from each group.

$(72m^2 - 72) - 17m = 72(m^2 - 1) - 17m$

Step 4: Factor the Difference of Squares

The expression $m^2 - 1$ is a difference of squares, which can be factored as $(m - 1)(m + 1)$.

$72(m^2 - 1) - 17m = 72(m - 1)(m + 1) - 17m$

Step 5: Factor out the Common Factor

Finally, we factor out the common factor from the two terms.

$72(m - 1)(m + 1) - 17m = 17m(4m - 1) - 17m$

Step 6: Simplify the Expression

Simplifying the expression, we get:

$17m(4m - 1) - 17m = 17m(4m - 1 - 1)$

$17m(4m - 2)$

Step 7: Factor the Expression

Factoring the expression, we get:

$17m(4m - 2) = 17m(2(2m - 1))$

$17m(2)(2m - 1)$

$17(2m)(2m - 1)$

$34m(2m - 1)$

Conclusion

In conclusion, the expression $72m^2 - 17m - 72$ can be factored as $34m(2m - 1)$. This is the final answer to the problem.

Final Answer

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Q&A: Factoring Trinomials

Q: What is a trinomial?

A: A trinomial is a polynomial expression with three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are the different techniques used to factor trinomials?

A: There are several techniques used to factor trinomials, including:

• Greatest Common Factor (GCF) Method: This method involves finding the greatest common factor of the three terms in the trinomial and factoring it out.
• Grouping Method: This method involves grouping the first two terms and the last two terms of the trinomial and factoring out the common factors.
• Factoring by Grouping Method: This method involves grouping the terms in the trinomial and factoring out the common factors.

Q: How do I factor a trinomial using the GCF method?

A: To factor a trinomial using the GCF method, follow these steps:

1. Find the greatest common factor (GCF) of the three terms in the trinomial.
2. Factor out the GCF from each term.
3. Write the factored form of the trinomial.

Q: How do I factor a trinomial using the grouping method?

A: To factor a trinomial using the grouping method, follow these steps:

1. Group the first two terms and the last two terms of the trinomial.
2. Factor out the common factors from each group.
3. Write the factored form of the trinomial.

Q: How do I factor a trinomial using the factoring by grouping method?

A: To factor a trinomial using the factoring by grouping method, follow these steps:

1. Group the terms in the trinomial.
2. Factor out the common factors from each group.
3. Write the factored form of the trinomial.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

$a^2 - b^2 = (a - b)(a + b)$

Q: How do I factor a difference of squares?

A: To factor a difference of squares, follow these steps:

1. Identify the difference of squares.
2. Apply the difference of squares formula.
3. Write the factored form of the expression.

Q: What is the final answer to the problem $72m^2 - 17m - 72$?

A: The final answer to the problem $72m^2 - 17m - 72$ is:

$34m(2m - 1)$

Conclusion

In conclusion, factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. There are several techniques used to factor trinomials, including the GCF method, the grouping method, and the factoring by grouping method. By understanding these techniques and strategies, you can factor trinomials with ease.

Final Answer

The final answer is: $\boxed{34m(2m - 1)}$