Greatest Common Factor (GCF) Quiz

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