A common monomial factor is the largest monomial that divides two or more monomials evenly. It is also known as the greatest common factor (GCF) or highest common factor of the monomials involved
. To find the common monomial factor, you:
- Factor each monomial into its prime factors, including the numeric coefficients and variables with their powers.
- Identify the factors common to all monomials.
- Take the lowest exponent for each common variable.
- Multiply these common factors together to get the GCF monomial.
For example, for the monomials 12x3y212x^3y^212x3y2 and 18x2y418x^2y^418x2y4:
- Factor each: 12x3y2=22×3×x3×y212x^3y^2=2^2\times 3\times x^3\times y^212x3y2=22×3×x3×y2, and 18x2y4=2×32×x2×y418x^2y^4=2\times 3^2\times x^2\times y^418x2y4=2×32×x2×y4.
- Common factors are 2,3,x,y2,3,x,y2,3,x,y.
- Take the smallest powers: 21,31,x2,y22^1,3^1,x^2,y^221,31,x2,y2.
- Multiply: 2×3×x2×y2=6x2y22\times 3\times x^2\times y^2=6x^2y^22×3×x2×y2=6x2y2.
Thus, the common monomial factor is 6x2y26x^2y^26x2y2
. In general, the common monomial factor is a monomial that can be factored out of each term of a polynomial, simplifying the expression. It includes the greatest common divisor of the coefficients and the variables raised to the lowest powers present in all terms
. This process is fundamental in factoring polynomials and simplifying algebraic expressions
