Greatest Common Factor (GCF) Quiz
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Questions and Answers

What is the GCF of 9x^2 and 12x^3?

  • 3x
  • 3x^3
  • 3x^2 (correct)
  • 3x^4

    • How do you find the GCF of two terms?

  • By listing the prime factors of each term (correct)
  • By multiplying the two terms
  • By adding the two terms
  • By subtracting the two terms

    • What is the purpose of finding the GCF in factoring polynomials?

  • To subtract the terms from each other
  • To rewrite the polynomial as a product of polynomials of smaller degree (correct)
  • To multiply the terms together
  • To add the terms together

    • What is the GCF of 12a and 18a^2?

    <p>6a</p> Signup and view all the answers

    What is the factored form of 6x^2 + 9x?

    <p>3x(2x + 3)</p> Signup and view all the answers

    What is the GCF of 8x^2y and 12xy^2?

    <p>4xy</p> Signup and view all the answers

    What is the definition of a perfect square?

    <p>A number that can be expressed as a product of a number multiplied by itself</p> Signup and view all the answers

    How can the number 144 be rewritten in exponential form?

    <p>12²</p> Signup and view all the answers

    What is the formula for the difference of two squares?

    <p>x² - y² = (x + y) × (x - y)</p> Signup and view all the answers

    What is the importance of remembering the sum and difference of terms when factoring?

    <p>To ensure the correct factoring of the expression</p> Signup and view all the answers

    What is the purpose of rewriting an expression in exponential form?

    <p>To identify the perfect square</p> Signup and view all the answers

    What is the definition of a difference of two squares?

    <p>x² - y² = (x + y) × (x - y)</p> Signup and view all the answers

    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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    Test your understanding of the Greatest Common Factor, including how to find it using prime factorization and identifying common factors.

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