Factoring Greatest Common Monomial Factor

Summary

In this instructional video, the speaker explains the process of factoring polynomials using the Greatest Common Monomial Factor (GCMF) method. They begin by introducing the different types of polynomials—monomial, binomial, trinomial, and multinomial—and then proceed with detailed examples on how to identify the GCF of numerical coefficients and variables. Through clear steps, the speaker demonstrates how to simplify expressions by factoring out the GCF and dividing each term accordingly. This video is designed to provide a thorough understanding of GCMF and help viewers grasp the fundamentals of polynomial factoring.

Takeaways

• 😀 Factoring polynomials involves different techniques like GCMF, OTS, SDTC, PST, and GT/Qt, each applicable to different polynomial forms.
• 😀 There are four types of polynomials: monomial (one term), binomial (two terms), trinomial (three terms), and multinomial (four or more terms).
• 😀 The first factoring method, GCMF (Greatest Common Monomial Factor), involves identifying the greatest common factor of both the numerical coefficients and variables.
• 😀 To apply GCMF, start by finding the GCF of the numerical coefficients of the terms in the polynomial.
• 😀 After finding the GCF of the numerical coefficients, identify the variable with the smallest exponent across the terms, and use it as part of the GCF.
• 😀 The factored form of a polynomial can be written as the product of the GCMF and the quotient of the division of each term by the GCMF.
• 😀 In the example 3x² - 6x, the GCMF is 3x, and the factored form is 3x(x - 2).
• 😀 When factoring a trinomial like 27y³ + 9y² - 18y, the process is the same: first find the GCF, then divide each term by the GCF.
• 😀 For the trinomial example, the GCF is 9y, and the factored form is 9y(3y² + y - 2).
• 😀 Always factor out the greatest common factor from all terms of the polynomial first, before proceeding with further factorization techniques like difference of squares or sum/difference of cubes.
• 😀 Understanding the relationship between the terms and recognizing the polynomial type helps in selecting the correct factoring method and simplifies the process.

Q & A

What is factoring in polynomials?

Factoring polynomials is the reverse of multiplying terms. It involves expressing a polynomial as a product of its factors.

What are the different types of polynomial factoring techniques discussed in the script?

The five techniques discussed are: GCMF (Greatest Common Monomial Factor), OTS (Difference of Two Squares), SDTC (Sum or Difference of Two Cubes), PST (Perfect Square Trinomial), and GT or QT (General Trinomials or Quadratic Trinomials).

What are the four types of polynomials mentioned in the script?

The four types of polynomials are: Monomial (one term), Binomial (two terms), Trinomial (three terms), and Multinomial (four or more terms).

How do you factor a binomial like 3x² - 6x using the GCMF method?

First, find the GCF of the numerical coefficients (3 and -6), which is 3. Then, determine the variable with the lowest exponent (x). The GCF is 3x. Next, divide each term by 3x to get the second factor: (x - 2). The factored form is 3x(x - 2).

In the example 27y³ + 9y² - 18y, how do you determine the GCF?

The GCF of the numerical coefficients 27, 9, and -18 is 9. The common variable is y, with the lowest exponent being y. Therefore, the GCF is 9y.

For the polynomial 5n² + 15n, how do you factor it using the GCMF method?

The GCF of 5 and 15 is 5, and the common variable is n. The GCF is 5n. Dividing each term gives the second factor (n + 3), resulting in the factored form: 5n(n + 3).

What is the first step in factoring a trinomial like ta³b² + 6a²b⁴ + 12ab⁶?

First, find the GCF of the numerical coefficients 3, 6, and 12, which is 3. Then, identify the common variables. The smallest exponent for a is a¹, and for b, it's b². Therefore, the GCF is 3ab².

In the trinomial example ta³b² + 6a²b⁴ + 12ab⁶, how do you calculate the second factor after finding the GCF?

After factoring out 3ab², divide each term by the GCF: - ta³b² ÷ 3ab² = a² - 6a²b⁴ ÷ 3ab² = 2ab² - 12ab⁶ ÷ 3ab² = 4b⁴ The factored form is 3ab²(a² + 2ab² + 4b⁴).

Why is factoring important in algebra?

Factoring is crucial in algebra because it helps simplify expressions, solve equations, and understand the underlying structure of polynomials.

What do you do after determining the GCF of the polynomial's numerical coefficients and variables?

After finding the GCF, divide each term of the polynomial by the GCF to get the second factor (quotient). This process simplifies the polynomial and expresses it in its factored form.