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Questions and Answers
What is the first step in factoring a polynomial using the Greatest Common Monomial Factor?
What is the first step in factoring a polynomial using the Greatest Common Monomial Factor?
- Applying the distributive property
- Rearranging the terms in descending order
- Identifying the highest degree term
- Finding the GCMF of the coefficients and variables (correct)
If the polynomial is $6x^4 + 9x^3 - 3x^2$, what is the GCMF?
If the polynomial is $6x^4 + 9x^3 - 3x^2$, what is the GCMF?
- $6x^3$
- $9x^2$
- $3x^4$
- $3x^2$ (correct)
After factoring out the GCMF from the polynomial $10y^3 + 15y^2 + 5y$, what will be the resulting expression?
After factoring out the GCMF from the polynomial $10y^3 + 15y^2 + 5y$, what will be the resulting expression?
- $5y(2y^2 + 3y + 5)$
- $5y(2y^2 + 3y)$
- $5y(2y + 3 + 1/y)$
- $5y(2y^2 + 3y + 1)$ (correct)
Which of the following polynomials cannot be factored by GCMF?
Which of the following polynomials cannot be factored by GCMF?
When factoring the expression $20ab^2c - 15a^2bc^2$, what is the greatest common monomial factor?
When factoring the expression $20ab^2c - 15a^2bc^2$, what is the greatest common monomial factor?
What is the common variable factor for the terms $x^2y^2$ and $x^3y^2$?
What is the common variable factor for the terms $x^2y^2$ and $x^3y^2$?
Which of the following is the common variable factor of the terms $b^3c^2$ and $a^2c^3$?
Which of the following is the common variable factor of the terms $b^3c^2$ and $a^2c^3$?
Identify the common variable factor among the terms $x^2y^2z$, $x^3yz^2$, and $x^2y^2$.
Identify the common variable factor among the terms $x^2y^2z$, $x^3yz^2$, and $x^2y^2$.
For the terms $x^2y^2$ and $x^3y^2$, how can you express their common variable factor?
For the terms $x^2y^2$ and $x^3y^2$, how can you express their common variable factor?
Which common variable factor can be derived from the terms $b^3c^2$ and $a^2c^3$?
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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