Questions and Answers
What is the first step in factoring a polynomial using the Greatest Common Monomial Factor?
If the polynomial is $6x^4 + 9x^3 - 3x^2$, what is the GCMF?
After factoring out the GCMF from the polynomial $10y^3 + 15y^2 + 5y$, what will be the resulting expression?
Which of the following polynomials cannot be factored by GCMF?
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When factoring the expression $20ab^2c - 15a^2bc^2$, what is the greatest common monomial factor?
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What is the common variable factor for the terms $x^2y^2$ and $x^3y^2$?
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Which of the following is the common variable factor of the terms $b^3c^2$ and $a^2c^3$?
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Identify the common variable factor among the terms $x^2y^2z$, $x^3yz^2$, and $x^2y^2$.
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For the terms $x^2y^2$ and $x^3y^2$, how can you express their common variable factor?
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Which common variable factor can be derived from the terms $b^3c^2$ and $a^2c^3$?
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Study Notes
Factoring by Greatest Common Monomial Factor (GCMF)
- Identify common variable factors among terms by comparing variables and exponents.
- Examples of common variable factors include:
- For terms x²y² and x³y², the common factor is x²y².
- For terms b³c² and a²c³, the common factor is c².
- For terms x²y²z, x³yz², and x²y², the common factor is x²y.
Activity: Finding GCMF of Polynomials
- GCMF can be determined by factoring out the greatest common factor from each polynomial.
- To be filled for:
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- x² + 2x
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- 5x² - 10x³
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- 25x²y³ + 55xy³
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- 10c³ - 80c⁵ - 5c⁶ + 5c⁷
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- 12m⁵n² - 6m²n³ + 3mn.
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Possible Factors of Numbers/Expressions
- Analyze given expressions to find their factors:
- 8 has factors 4 and 2.
- 2x includes factors 2 and x.
- 5ab consists of factors 5, a, and b.
- 12z can be factored as 4, 3, z or 6, 2, z.
- 20xy can be factored as 2, 10, x, y or 4, 5, x, y.
Greatest Common Factor (GCF)
- The GCF is defined as the largest number that divides all members of a set.
- Examples of GCF for different sets include:
- For 4 and 12, the GCF is 4.
- For 9 and 15, the GCF is 3.
- For 18 and 24, the GCF is 6.
- For 22, 33, and 44, the GCF is 11.
Understanding Factors
- Factors are numbers or expressions multiplied together to yield a product.
- Example: In 2 × 3 = 6, both 2 and 3 are factors of 6.
- Factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24.
Steps to Find GCMF of a Polynomial
- Determine the GCF of all coefficients from the terms of the polynomial.
- Identify the common variable factor based on the lowest exponent of variables present in each term.
- Multiply the GCF of coefficients with the common variable factor to calculate the overall GCF.
Common Variable Factor
- The common variable factor represents the variable shared among terms at its lowest exponent.
- Examples include:
- For terms x and x², the common variable factor is x.
- For terms ab and a³b², the common variable factor is ab.
Factoring Polynomials using GCMF
- Identify the GCMF present in each term of the polynomial.
- Express each term as a product of the GCMF and another factor.
- Apply the distributive property to factor out the GCMF.
- Illustrate the expression in its factored form for clarity.
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Description
This quiz focuses on the first lesson about factoring using the Greatest Common Monomial Factor (GCMF). Test your understanding of how to identify and apply GCMF in various mathematical expressions. Perfect for students preparing for algebra assessments.
