Factor This Trinomial: $x^2 - 7x + 10$A. \[$(x-1)(x-10)\$\]B. \[$(x-2)(x+5)\$\]C. \[$(x-2)(x-5)\$\]D. \[$(x-1)(x+10)\$\]

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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 explore the process of factoring trinomials and apply it to the given trinomial x2−7x+10x^2 - 7x + 10. We will also discuss the different methods of factoring trinomials and provide examples to illustrate the concepts.

What is a Trinomial?

A trinomial is a quadratic expression that consists of three terms. It can be written in the form ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants. The trinomial x2−7x+10x^2 - 7x + 10 is a specific example of a trinomial, where a=1a = 1, b=−7b = -7, and c=10c = 10.

Methods of Factoring Trinomials

There are several methods of factoring trinomials, including:

  • Factoring by Grouping: This method involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.
  • Factoring by Using the AC Method: This method involves using the product of the coefficients of the quadratic and linear terms to find the factors of the trinomial.
  • Factoring by Using the Coefficient Method: This method involves using the coefficients of the trinomial to find the factors.

Factoring the Trinomial x2−7x+10x^2 - 7x + 10

To factor the trinomial x2−7x+10x^2 - 7x + 10, we can use the method of factoring by grouping. This involves grouping the terms of the trinomial into two pairs and then factoring out the GCF from each pair.

Step 1: Group the Terms

The trinomial x2−7x+10x^2 - 7x + 10 can be grouped into two pairs as follows:

  • (x2−7x)(x^2 - 7x)
  • (10)(10)

Step 2: Factor Out the GCF

The GCF of the first pair is xx, and the GCF of the second pair is 1010. We can factor out the GCF from each pair as follows:

  • (x(x−7))(x(x - 7))
  • (10)(10)

Step 3: Combine the Factors

We can combine the factors from each pair to get the final factorization of the trinomial:

(x−7)(x−10)(x - 7)(x - 10)

Conclusion

In this article, we have explored the process of factoring trinomials and applied it to the given trinomial x2−7x+10x^2 - 7x + 10. We have also discussed the different methods of factoring trinomials and provided examples to illustrate the concepts. By following the steps outlined in this article, you should be able to factor trinomials with ease.

Answer

The correct answer is:

A. {(x-1)(x-10)$}$

Why is this the correct answer?

The correct answer is A. {(x-1)(x-10)$}$ because the trinomial x2−7x+10x^2 - 7x + 10 can be factored as (x−1)(x−10)(x - 1)(x - 10). This is because the GCF of the trinomial is 11, and the factors of the trinomial are (x−1)(x - 1) and (x−10)(x - 10).

Comparison of Options

Let's compare the options to see why A. {(x-1)(x-10)$}$ is the correct answer:

  • Option A: (x−1)(x−10)(x - 1)(x - 10)
  • Option B: (x−2)(x+5)(x - 2)(x + 5)
  • Option C: (x−2)(x−5)(x - 2)(x - 5)
  • Option D: (x−1)(x+10)(x - 1)(x + 10)

The correct answer is A. {(x-1)(x-10)$}$ because it is the only option that matches the factorization of the trinomial x2−7x+10x^2 - 7x + 10.

Conclusion

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 provide a Q&A guide to help you understand the process of factoring trinomials and address common questions and concerns.

Q: What is a trinomial?

A: A trinomial is a quadratic expression that consists of three terms. It can be written in the form ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants.

Q: What are the different methods of factoring trinomials?

A: There are several methods of factoring trinomials, including:

  • Factoring by Grouping: This method involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.
  • Factoring by Using the AC Method: This method involves using the product of the coefficients of the quadratic and linear terms to find the factors of the trinomial.
  • Factoring by Using the Coefficient Method: This method involves using the coefficients of the trinomial to find the factors.

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 terms of the trinomial into two pairs.
  2. Factor out the GCF from each pair.
  3. Combine the factors from each pair to get the final factorization of the trinomial.

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

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

  1. Find the product of the coefficients of the quadratic and linear terms.
  2. Find the factors of the product that add up to the coefficient of the linear term.
  3. Write the factors as a product of two binomials.

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

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

  1. Find the factors of the constant term.
  2. Find the factors of the coefficient of the linear term.
  3. Write the factors as a product of two binomials.

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

A: Some common mistakes to avoid when factoring trinomials include:

  • Not grouping the terms correctly: Make sure to group the terms of the trinomial into two pairs.
  • Not factoring out the GCF correctly: Make sure to factor out the GCF from each pair.
  • Not combining the factors correctly: Make sure to combine the factors from each pair to get the final factorization of the trinomial.

Q: How do I check my answer when factoring a trinomial?

A: To check your answer when factoring a trinomial, follow these steps:

  1. Multiply the factors together to get the original trinomial.
  2. Simplify the expression to make sure it matches the original trinomial.

Conclusion

In conclusion, factoring trinomials is an important concept in algebra that involves expressing a quadratic expression as a product of two binomials. By following the steps outlined in this article and avoiding common mistakes, you should be able to factor trinomials with ease. Remember to check your answer by multiplying the factors together and simplifying the expression.

Additional Resources

For more information on factoring trinomials, check out the following resources:

  • Algebra textbooks: Many algebra textbooks include chapters on factoring trinomials.
  • Online resources: Websites such as Khan Academy and Mathway offer video lessons and interactive exercises on factoring trinomials.
  • Practice problems: Practice factoring trinomials with online resources such as IXL and Mathway.

Final Thoughts

Factoring trinomials is a fundamental concept in algebra that requires practice and patience. By following the steps outlined in this article and avoiding common mistakes, you should be able to factor trinomials with ease. Remember to check your answer by multiplying the factors together and simplifying the expression. Good luck!