Mastering Factoring by Grouping: A PreCalculus Approach

PreAlgebra Grades High School 1:25 Video
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Lesson Description

Learn the factoring by grouping method for polynomials with four terms. This lesson covers identifying when to use this technique, performing the steps, and checking your work.

Video Resource

Factoring by Grouping (4 terms)

Mario's Math Tutoring

Duration: 1:25
Watch on YouTube

Key Concepts

  • Polynomial expressions with four terms
  • Greatest Common Factor (GCF)
  • Binomial factoring

Learning Objectives

  • Identify when factoring by grouping is the appropriate method.
  • Correctly factor polynomial expressions with four terms using the grouping method.
  • Verify the factored form of a polynomial expression.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of factoring and its importance in solving polynomial equations. Introduce the specific scenario where factoring by grouping is applicable: polynomial expressions with four terms. Briefly mention that this method relies on identifying common factors within groups of terms.
  • Identifying When to Use Factoring by Grouping (2 mins)
    Emphasize that factoring by grouping is primarily used when dealing with polynomial expressions containing four terms. Highlight the fact that terms are separated by addition or subtraction signs. This serves as the initial indicator to consider this method.
  • Grouping and Factoring GCF (10 mins)
    Demonstrate the process of grouping the first two terms and the last two terms of the polynomial expression. For subtraction, clarify rewriting as addition of a negative. Illustrate how to factor out the greatest common factor (GCF) from each group separately. Provide step-by-step examples to ensure clarity. Emphasize the importance of identifying common binomial factors after factoring out the GCF.
  • Factoring out the Common Binomial (8 mins)
    Explain the process of factoring out the common binomial factor from the two groups. Reinforce that this is essentially the reverse of the distributive property. Illustrate how, after factoring out the common binomial, the remaining terms form the second factor. Provide clear examples to solidify understanding.
  • Checking Your Work (5 mins)
    Explain how to check the work by re-multiplying the factored polynomials using the distributive property, also known as FOIL (First, Outer, Inner, Last). The resultant polynomial should match the original polynomial before it was factored.

Interactive Exercises

  • Factoring Practice
    Provide students with a worksheet containing various polynomial expressions with four terms. Have them practice factoring these expressions using the grouping method. Encourage them to check their work by re-multiplying the factored polynomials to ensure they arrive at the original expression. This hands-on activity will help reinforce the concepts learned and build confidence in their factoring abilities.

Discussion Questions

  • Why is it important to understand factoring techniques in precalculus?
  • What are some common mistakes to avoid when factoring by grouping?

Skills Developed

  • Factoring Polynomials
  • Algebraic Manipulation

Multiple Choice Questions

Question 1:

When is factoring by grouping most likely to be an effective technique?

Correct Answer: When there are four terms

Question 2:

What is the first step in factoring by grouping?

Correct Answer: Group the first two terms and the last two terms.

Question 3:

After grouping, what is the next step?

Correct Answer: Factor out the greatest common factor from each group.

Question 4:

What should you look for after factoring out the GCF from each group?

Correct Answer: A common binomial factor.

Question 5:

If (x+2) is a common binomial factor, what do you do with it?

Correct Answer: Factor it out of both groups.

Question 6:

Which of the following is a factored form of x³ + 2x² + 3x + 6?

Correct Answer: (x²+3)(x+2)

Question 7:

When factoring by grouping, if the last two terms are ' - ax - ay' how should you group them initially?

Correct Answer: -(ax + ay)

Question 8:

What is the greatest common factor (GCF) of 4x³ + 8x²?

Correct Answer: 4x²

Question 9:

After factoring by grouping, how can you check if your factored expression is correct?

Correct Answer: Re-multiply the factored expressions.

Question 10:

What happens if, after grouping and factoring out the GCF, there is no common binomial factor?

Correct Answer: Try a different grouping or method.

Fill in the Blank Questions

Question 1:

Factoring by grouping is often used with polynomials that have ____ terms.

Correct Answer: four

Question 2:

The first step in factoring by grouping is to _____ the terms.

Correct Answer: group

Question 3:

After grouping, you factor out the _____ from each group.

Correct Answer: GCF

Question 4:

GCF stands for ______.

Correct Answer: Greatest Common Factor

Question 5:

If there is subtraction in the original polynomial it may be rewritten as adding a ____.

Correct Answer: negative

Question 6:

After factoring the GCF, you should identify the common ______ factor.

Correct Answer: binomial

Question 7:

Factoring is the reverse process of the ______ property.

Correct Answer: distributive

Question 8:

If you cannot factor out a GCF from both groupings, you should try a ______ grouping method.

Correct Answer: different

Question 9:

To verify your work, you can ______ the factored expressions.

Correct Answer: multiply

Question 10:

If after factoring out the GCF from both groups, you are left with nothing, then you need to put a ______ as a placeholder.

Correct Answer: 1

Educational Standards

PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation.

Teaching Materials

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