- 1). Pinpoint which variables appear in every term of the polynomial. In the polynomial 6x^2y + 12x^4y^2 + 3 x^2yz, for example, x and y appear in every term, so they are both part of the GCF. Z appears in only one term, so do not include it in the GCF.
- 2). Use each variable with the lowest exponent it has in the polynomial for the GCF. In 6x^2y + 12x^4y^2 + 3 x^2yz, 2 is the lowest exponent with x, so use x^2 in the GCF. The lowest exponent of y is 1 -- because y equals y^1 -- so use y in the GCF. This means x^2y is part of the GCF for this polynomial.
- 3). Factor the coefficient of each term in the polynomial using prime factorization to determine the GCF of the coefficients. For example, the coefficients of 6x^2y + 12x^4y^2 + 3 x^2yz are 6, 12, and 3. The prime factorizations are 2*3, 2*2*3 and 3, respectively. Since only 3 appears in every factorization, it is the GCF (see Resource).
- 4). Write the GCF of the coefficients before the GCF of the variables to get the complete GCF for the polynomial. For example, the GCF of 6x^2y + 12x^4y^2 + 3 x^2yz is 3x^2y.
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