Common Monomial Factoring

Introduction

• Topic: Common monomial factoring, a key concept in mathematics.
• Importance: Understanding the greatest common factor (GCF) in expressions.

Key Concepts

1. Finding the GCF
   • Identify the GCF for coefficients and variables in the terms.
   • Compare the coefficients and choose the highest number that divides both.
   • For variables, compare the exponents and choose the variable with the least exponent.

Example 1: 6xy^3 + 24x^2y

• Step 1: Identify GCF

  • Coefficients: GCF of 6 and 24 is 6.
  • Variables:
    • x (from x and x^2)
    • y (from y^3 and y)
  • GCF = 6xy.

• Step 2: Divide each term by GCF

  • First term:
    • (6xy^3) / (6xy) = y^2
  • Second term:
    • (24x^2y) / (6xy) = 4x

• Result: Factors are 6xy( y^2 + 4x ).

Example 2: 36x^3y^6 - 63x^5y^7

• Step 1: Identify GCF

  • Coefficients: GCF of 36 and -63 is 9.
  • Variables:
    • x^3 (from x^3 and x^5)
    • y^6 (from y^6 and y^7)
  • GCF = 9x^3y^6.

• Step 2: Divide each term by GCF

  • First term:
    • (36x^3y^6) / (9x^3y^6) = 4
  • Second term:
    • (-63x^5y^7) / (9x^3y^6) = -7x^2y

• Result: Factors are 9x^3y^6( 4 - 7x^2y ).

Example 3: 2x^2 - 4x^3 + 18x^5

• Step 1: Identify GCF

  • Coefficients: GCF of 2, -4, and 18 is 2.
  • Variables:
    • x^2 (from x^2, x^3, and x^5)
  • GCF = 2x^2.

• Step 2: Divide each term by GCF

  • First term:
    • (2x^2) / (2x^2) = 1
  • Second term:
    • (-4x^3) / (2x^2) = -2x
  • Third term:
    • (18x^5) / (2x^2) = 9x^3

• Result: Factors are 2x^2( 1 - 2x + 9x^3 ).

Example 4: 4x(x + 3) - 1(x + 3)

• Step 1: Identify common binomial

  • Common binomial: (x + 3).

• Step 2: Factor out the common binomial

  • Factorization gives: (x + 3)(4x - 1).

Conclusion

• Review of common monomial factoring.
• Encouragement to practice and check answers using the distributive property.
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