Mastering Factoring Trinomials: Unlocking Common Factors

Algebra 2 Grades High School 8:47 Video
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Lesson Description

Learn how to factor trinomials by first identifying and extracting common factors, then applying advanced factoring techniques for trinomials with lead coefficients. This lesson provides a step-by-step guide to simplifying complex expressions.

Video Resource

Factoring Trinomials with Common Factor

Kevinmathscience

Duration: 8:47
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Key Concepts

  • Identifying and extracting common factors from trinomials.
  • Factoring trinomials with a lead coefficient (coefficient other than 1).
  • Using the 'lines' technique to factor trinomials where the leading coefficient is not 1.

Learning Objectives

  • Students will be able to identify and factor out the greatest common factor (GCF) from a trinomial.
  • Students will be able to factor trinomials with a leading coefficient after extracting the GCF.
  • Students will be able to apply a structured method (like the 'lines' technique) to factor complex trinomials.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of factoring and greatest common factors. Briefly discuss why factoring is important in algebra (simplifying expressions, solving equations).
  • Identifying Common Factors (10 mins)
    Explain and demonstrate how to identify the greatest common factor (GCF) in a trinomial. Work through examples where both numerical and variable common factors are present. Emphasize the importance of extracting the *greatest* common factor.
  • Factoring Trinomials with Lead Coefficients (20 mins)
    Introduce the 'lines' technique (or another preferred method) for factoring trinomials where the coefficient of the x² term is not 1. Break down the steps involved, emphasizing the importance of finding the correct factor pairs and signs. Use examples from the video.
  • Practice Problems (15 mins)
    Students work through practice problems, applying the techniques learned. Provide a variety of problems with increasing difficulty. Circulate and provide assistance as needed. Encourage students to check their work by expanding the factored expression.
  • Review and Wrap-up (5 mins)
    Review the key concepts and steps involved in factoring trinomials with common factors. Answer any remaining questions. Preview upcoming topics (e.g., special factoring patterns).

Interactive Exercises

  • Factor the Trinomial
    Present students with a series of trinomials. They must first identify and factor out the GCF, then factor the resulting trinomial using an appropriate method. (Example: 6x² + 15x + 9)
  • Error Analysis
    Present students with incorrectly factored trinomials. They must identify the error and correct it. (Example: Student factors 4x² + 8x + 4 as 2(2x² + 4x + 2). Error: Did not factor out the *greatest* common factor.)

Discussion Questions

  • Why is it important to find the *greatest* common factor, rather than just *a* common factor?
  • Can you explain in your own words the 'lines' technique for factoring trinomials with a lead coefficient?
  • How can you check your answer after factoring a trinomial?

Skills Developed

  • Factoring Polynomials
  • Problem-Solving
  • Analytical Thinking
  • Attention to Detail

Multiple Choice Questions

Question 1:

What is the first step in factoring the trinomial 4x² + 8x + 12?

Correct Answer: Factor out a 4

Question 2:

Which of the following is the greatest common factor of 9x³ - 18x² + 27x?

Correct Answer: 9x

Question 3:

After factoring out the GCF, what remains when factoring 2x² - 6x + 4?

Correct Answer: x² - 3x + 2

Question 4:

Which expression represents the completely factored form of 3x^2 + 12x + 9?

Correct Answer: 3(x+1)(x+3)

Question 5:

If you factor out 5x from 10x^3 - 15x^2 + 25x, what trinomial remains?

Correct Answer: 2x^2 - 3x + 5

Question 6:

When factoring a trinomial with a lead coefficient after removing the GCF, what method can be useful?

Correct Answer: The 'lines' technique

Question 7:

Factoring out the GCF will always make the problem _____.

Correct Answer: Easier

Question 8:

If a trinomial can't be factored after extracting the GCF, it is a ____ polynomial.

Correct Answer: Prime

Question 9:

After factoring, how do you ensure the answer is correct?

Correct Answer: Expand the expression

Question 10:

What is the completely factored form of 2x^2 + 8x + 6?

Correct Answer: 2(x+1)(x+3)

Fill in the Blank Questions

Question 1:

The first step when factoring any trinomial is to look for a ___________.

Correct Answer: GCF

Question 2:

The 'lines' technique is helpful when factoring trinomials with a ___________ other than 1.

Correct Answer: lead coefficient

Question 3:

If a polynomial cannot be factored, it is called a ___________ polynomial.

Correct Answer: prime

Question 4:

The acronym GCF stands for ___________.

Correct Answer: greatest common factor

Question 5:

After factoring a polynomial, you can ___________ the result to check if it's correct.

Correct Answer: expand

Question 6:

Finding the GCF involves identifying numbers and _______ that divide evenly into all terms of the trinomial.

Correct Answer: variables

Question 7:

Factoring a trinomial reverses the process of __________ two binomials.

Correct Answer: multiplying

Question 8:

If 5 is the GCF of a trinomial, all coefficients will be divisible by ___________.

Correct Answer: 5

Question 9:

The opposite of expanding is _______.

Correct Answer: factoring

Question 10:

The expression inside parenthesis after the GCF has been factored is a ___________.

Correct Answer: polynomial

Educational Standards

A2.A.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies. A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

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