6.1 Greatest Common Factor and Factor by Grouping --Lesson 20--FA26
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Overview
Rachel Soyk teaches how to find greatest common factors and factor polynomials by reversing the distributive property. The lesson connects prime factorization and the cake method to extracting monomial or binomial GCFs, then applies grouping to four-term polynomials.
Key takeaways
- For a polynomial GCF, take the greatest common factor of the numerical coefficients and the lowest exponent of each variable shared by every term; for 21x³, 9x², and 15x, this gives 3x.
- After extracting a GCF, divide each term by that factor to construct the remaining polynomial, as in 8m³ − 12m²n + 20mn² = 4m(2m² − 3mn + 5n²).
- When every term begins with a negative coefficient, factoring out a negative GCF makes the leading term inside the parentheses positive.
- A repeated binomial can itself be the GCF: 3y(y + 7) − 4(y + 7) factors directly to (y + 7)(3y − 4).
- For a four-term polynomial, factor each pair separately and check that the remaining binomials match; matching factors can then be extracted to complete the grouping.
- FOIL provides a verification method for factored expressions, while addition's commutative property means expanded terms may be reordered without changing the result.
Chapters
- Factors 56 into primes as 2 × 2 × 2 × 7, or 2³ × 7, and distinguishes listing primes from writing a prime factorization.
- Reviews the cake method for finding the LCM of 18 and 24: divide both by 6, then multiply 6 × 3 × 4 to get 72.
- Introduces three goals: finding GCFs of expressions, factoring out a monomial GCF, and factoring by grouping.
- For 21x³, 9x², and 15x, the coefficients share a factor of 3 and the smallest shared x exponent is 1, so the GCF is 3x.
- Soyk explains that the GCF of variable terms uses the smallest exponent present in every term.
- The cake method's diagonal product gives the GCF, while multiplying the full border gives the LCM.
- Factors 8m³ − 12m²n + 20mn² by extracting 4m, leaving 2m² − 3mn + 5n² in parentheses.
- Uses the numerical GCF of coefficients and the lowest shared variable exponents to identify a polynomial's common factor.
- Shows quick cases such as 5x³ − 25x² = 5x²(x − 5), and emphasizes factoring out a negative GCF when the leading term is negative.
- Factors 3y(y + 7) − 4(y + 7) by recognizing y + 7 as the common binomial, giving (y + 7)(3y − 4).
- Soyk recommends checking for shared factors before distributing or combining terms, since recognizing the pattern avoids unnecessary quadratic factoring.
- Introduces the grouping strategy for polynomials with four terms when no single factor is shared by all terms.
- Groups xy + 3y + 2x + 6 into two pairs, factors each pair to obtain y(x + 3) + 2(x + 3), then extracts x + 3 to get (x + 3)(y + 2).
- For a second grouping example, the first pair yields x(x + 3) and the second yields −2(x + 3), producing (x + 3)(x − 2).
- Factors 6x² − 3x − 4x + 2 by grouping: 3x(2x − 1) − 2(2x − 1) becomes (2x − 1)(3x − 2).
- Checks the first result by FOIL and notes that the expanded terms can appear in a different order because addition is commutative.
Transcript excerpt
0:02 Right. Finally, we're on to chapter six. Being able to um find greatest common factors and factor by grouping. Now, in this uh section we're kind of bringing back our knowledge or method of using the cake method, which is kind of why I like that method so much because kind of kill two birds with one stone. You're able to identify what your LCM is, uh your least common multiple, also known as your least common denominator. And then in that same method, you can identify what your GCF is, uh what your greatest common factor is.
0:50 Three learning objectives in this section, we're going to learn how to find the GCF of two or more expressions. We're going to learn how to factor the GCF of a monomial and then last but not least, we are going to learn how to factor by grouping. Let's jump into some warm-up problems first. That first one you should feel very confident in at this point in time. Uh we want to factor 56 into primes. And so, think about uh doing your whole prime factorization and your factor tree.
1:38 So, we have 56 here. And let's say we break that up into 8 * what? 8 * what gives you 56? 7. And is 7 a prime number? It is, so we circle it. Now, let's look at 8. Is 8 a prime number? No, we can break that up into 4 and 2. Is 2 a prime number? Yep, so we're going to circle that. 4 isn't, so what can we break 4 into? 2 and 2, and 2 and 2 are both So,
2:27 if we were to write this in expanded form, we would have 2 * 2 * 2 * 7. In exponential form, we would have 2 to the
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Related notes
Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Rachel Soyk.