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Questions and Answers
What is the GCF of 9x^2 and 12x^3?
What is the GCF of 9x^2 and 12x^3?
How do you find the GCF of two terms?
How do you find the GCF of two terms?
What is the purpose of finding the GCF in factoring polynomials?
What is the purpose of finding the GCF in factoring polynomials?
What is the GCF of 12a and 18a^2?
What is the GCF of 12a and 18a^2?
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What is the factored form of 6x^2 + 9x?
What is the factored form of 6x^2 + 9x?
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What is the GCF of 8x^2y and 12xy^2?
What is the GCF of 8x^2y and 12xy^2?
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What is the definition of a perfect square?
What is the definition of a perfect square?
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How can the number 144 be rewritten in exponential form?
How can the number 144 be rewritten in exponential form?
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What is the formula for the difference of two squares?
What is the formula for the difference of two squares?
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What is the importance of remembering the sum and difference of terms when factoring?
What is the importance of remembering the sum and difference of terms when factoring?
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What is the purpose of rewriting an expression in exponential form?
What is the purpose of rewriting an expression in exponential form?
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What is the definition of a difference of two squares?
What is the definition of a difference of two squares?
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Study Notes
Greatest Common Factor (GCF)
- GCF refers to the common factor with the greatest numerical value and variables with the least degree
- To find the GCF, list the prime factors of each term and identify the common factors
- The GCF can be found by listing the prime factors or by using prime factorization
Finding GCF using Prime Factorization
- Break down each term into its prime factors
- Identify the common prime factors among all terms
- The GCF is the product of the common prime factors
Examples of Finding GCF
- Find the GCF of 6x^2 and 15x^4: 3x^2
- Find the GCF of 6a and 18aB: 6a
- Find the GCF of 10a and 12a^2B: 2a
- Find the GCF of -8x^2y and 16xy: 2xy
- Find the GCF of 8aB^3 and 10a^2B^2: 2aB^2
Factoring Polynomials using GCF
- To factor a polynomial, find the GCF and rewrite the polynomial as a product of polynomials of smaller degree
- Use the distributive property to rewrite the polynomial
- Factor out the GCF and rewrite the polynomial in factored form
Examples of Factoring Polynomials
- Factor 4x^2 + 6x: 2x(2x + 3)
- Factor 3x^2 + 6x: 3x(x + 2)
- Factor 6x^4 - 14x^2: 2x^2(3x^2 - 7)
Other Examples of Factoring Polynomials
- Factor 7/8a + 3 - Ca + 3: (7/8 - C)(a + 3)
- Factor 4(3B - 1) + Phi(B - 1) + 4C(3B - 1): (3B - 1)(4 + Phi + 4C)
Greatest Common Factor (GCF)
- GCF is the common factor with the greatest numerical value and variables with the least degree
- To find the GCF, list the prime factors of each term and identify the common factors
- GCF can be found using prime factors or prime factorization
Finding GCF
- Break down each term into its prime factors to find the GCF
- Identify the common prime factors among all terms
- GCF is the product of the common prime factors
Examples of GCF
- GCF of 6x^2 and 15x^4 is 3x^2
- GCF of 6a and 18aB is 6a
- GCF of 10a and 12a^2B is 2a
- GCF of -8x^2y and 16xy is 2xy
- GCF of 8aB^3 and 10a^2B^2 is 2aB^2
Factoring Polynomials using GCF
- Factor a polynomial by finding the GCF and rewriting it as a product of polynomials of smaller degree
- Use the distributive property to rewrite the polynomial
- Factor out the GCF and rewrite in factored form
Examples of Factoring Polynomials
- 4x^2 + 6x factors to 2x(2x + 3)
- 3x^2 + 6x factors to 3x(x + 2)
- 6x^4 - 14x^2 factors to 2x^2(3x^2 - 7)
- 7/8a + 3 - Ca + 3 factors to (7/8 - C)(a + 3)
- 4(3B - 1) + Phi(B - 1) + 4C(3B - 1) factors to (3B - 1)(4 + Phi + 4C)
Identifying Perfect Squares
- A perfect square is a number that can be expressed as a product of a number multiplied by itself
- Examples of perfect squares include 25, 81, 144, 100, 64, and 121
Rewriting Perfect Squares in Exponential Form
- Each perfect square can be rewritten in exponential form using the square root of the number
- Examples of exponential form include 5², 9², 12², 10², 8², and 11²
Difference of Two Squares
- The difference of two squares is a product of the sum and difference of two terms: x² - y² = (x + y) × (x - y)
- Examples of the difference of two squares include 9x² - 100 = (3x + 10) × (3x - 10) and 4x² - 81 = (2x + 9) × (2x - 9)
Factoring Difference of Two Squares
- To factor a difference of two squares, rewrite the expression as x² - y², then identify the square roots of x and y
- Examples of factoring the difference of two squares include 9x² - 100 = (3x + 10) × (3x - 10), 4x² - 81 = (2x + 9) × (2x - 9), and 81m² - 4n² = (9m + 2n) × (9m - 2n)
Important Notes
- When factoring, always remember to use the sum and difference of the terms
- Take note of exponents when rewriting expressions in exponential form
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Description
Test your understanding of the Greatest Common Factor, including how to find it using prime factorization and identifying common factors.
