Factoring High-Degree Trinomials: Beyond the Quadratic

Algebra 2 Grades High School 8:17 Video
Print Lesson Plan Watch Video

Lesson Description

Learn to factor trinomials with higher-degree terms using substitution and strategic factoring techniques, building upon your existing knowledge of quadratic factoring.

Video Resource

Factoring High Degree Trinomials

Kevinmathscience

Duration: 8:17
Watch on YouTube

Key Concepts

  • Recognizing trinomial patterns with higher-degree terms.
  • Using substitution to simplify factoring.
  • Strategic application of factoring techniques.
  • Identifying and extracting common factors as a first step.

Learning Objectives

  • Students will be able to identify trinomials with higher-degree terms that can be factored.
  • Students will be able to use substitution to simplify the factoring process.
  • Students will be able to apply appropriate factoring techniques to solve high-degree trinomials.
  • Students will be able to extract common factors prior to factoring a trinomial.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a trinomial (ax^2 + bx + c) and the basic factoring techniques. Briefly discuss how this knowledge can be extended to trinomials with higher powers, emphasizing that the key is to recognize the pattern: a(x^n)^2 + b(x^n) + c.
  • Video Presentation and Guided Notes (15 mins)
    Play the Kevinmathscience video on "Factoring High Degree Trinomials." Students should take notes on the examples presented, paying close attention to the substitution method and how the factoring techniques from quadratic trinomials are applied. Encourage students to pause the video and attempt the examples independently before watching the solution.
  • Practice Problems and Group Work (20 mins)
    Provide students with a set of practice problems involving high-degree trinomials. Have them work in small groups to solve these problems, encouraging them to discuss their strategies and help each other. Circulate the classroom to provide guidance and address any difficulties.
  • Discussion and Wrap-Up (10 mins)
    Bring the class together for a discussion of the practice problems. Have students share their solutions and explain their reasoning. Address any common misconceptions and summarize the key steps involved in factoring high-degree trinomials.

Interactive Exercises

  • Factoring Challenge
    Present a series of increasingly complex high-degree trinomials and challenge students to factor them correctly and quickly. Time them and create a leaderboard to foster a competitive and engaging environment.
  • Error Analysis
    Provide students with worked-out solutions to factoring problems, some of which contain errors. Have students identify and correct the errors, explaining the correct steps.

Discussion Questions

  • How does recognizing the pattern of a trinomial help in factoring high-degree expressions?
  • What are the advantages and disadvantages of using substitution when factoring these types of trinomials?
  • How does extracting a common factor simplify the overall factoring process?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Pattern recognition
  • Critical Thinking

Multiple Choice Questions

Question 1:

Which of the following expressions is a trinomial with a higher degree that can be factored using techniques similar to quadratic factoring?

Correct Answer: x^4 + 5x^2 + 6

Question 2:

What is the first step you should always take when factoring any polynomial, including high-degree trinomials?

Correct Answer: Check for a greatest common factor (GCF)

Question 3:

In the expression x^6 - 7x^3 + 12, what substitution could be used to simplify the factoring process?

Correct Answer: Let u = x^3

Question 4:

After substituting 'u' for x^3 in the trinomial x^6 - 7x^3 + 12, what is the resulting quadratic trinomial in terms of 'u'?

Correct Answer: u^2 - 7u + 12

Question 5:

If a high-degree trinomial factors into (x^2 - 3)(x^2 - 4), what are the possible real roots of the polynomial?

Correct Answer: ±√3, ±2

Question 6:

When factoring a trinomial like 2x^4 + 5x^2 + 2, after factoring, you must remember to:

Correct Answer: Substitute back the original variable

Question 7:

What is a potential issue to be aware of when solving high-degree polynomials after factoring?

Correct Answer: All of the above

Question 8:

Which of the following is NOT a valid factoring technique applicable to high-degree trinomials?

Correct Answer: Long Division

Question 9:

If a substitution simplifies a high-degree trinomial to u^2 - 9 = 0, what are the values of 'u'?

Correct Answer: ±3

Question 10:

Why is it important to check your solutions after factoring and solving high-degree trinomials?

Correct Answer: To identify extraneous solutions

Fill in the Blank Questions

Question 1:

A trinomial must have exactly ______ terms.

Correct Answer: three

Question 2:

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

Correct Answer: greatest common factor

Question 3:

If you substitute u = x^2 in the trinomial x^4 + 5x^2 + 4, the resulting trinomial is u^2 + 5u + _______

Correct Answer: 4

Question 4:

When solving a polynomial equation after factoring, each factor is set equal to _________ to find the solutions.

Correct Answer: zero

Question 5:

After factoring, it is essential to ____________ back to the original variable to express the solution correctly.

Correct Answer: substitute

Question 6:

If x^6 + 2x^3 + 1 is factored, it can be written as (x^3 + 1)^_______.

Correct Answer: 2

Question 7:

A solution that appears valid but does not satisfy the original equation is called an ____________ solution.

Correct Answer: extraneous

Question 8:

The difference of squares factoring pattern is a^2 - b^2 = (a + b)(_______).

Correct Answer: a - b

Question 9:

The expression x^8 - 16 can be seen as a difference of squares, specifically (x^4)^2 - _______^2.

Correct Answer: 4

Question 10:

The solutions to a factored polynomial equation are also known as the ________ of the polynomial.

Correct Answer: roots

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.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans