What is the greatest common factor of the three terms in the polynomial 12x^4y^2 + 18x²y^4 + 24x³y?
6x²y is the greatest common factor of the three terms in the polynomial.
To determine the greatest common factor (GCF) of the polynomial 12x^4y^2 + 18x²y^4 + 24x³y, we need to find the highest degree of each variable and the coefficients that can be factored out from all three terms. The GCF is 6x²y, which is common to all terms.
While 2xy is a common factor of the terms, it is not the greatest one. The coefficients of 12, 18, and 24 share a higher common factor of 6, and the variable parts can be factored more extensively as well.
This choice includes a common factor, but it fails to account for the highest common coefficients and variable degrees. The GCF should include a coefficient of 6 and higher powers of both x and y, making this choice insufficient.
This option is not a common factor as it exceeds the variable degrees present in all terms. Specifically, it has x raised to the fourth power and y raised to the fourth power, which are not present in all three terms of the polynomial.
This is the correct answer as it is the highest factor that can be factored out of all three terms: 12x^4y^2, 18x²y^4, and 24x³y. The coefficients share a GCF of 6, and the minimum power of x is x² and for y is y.
This choice is incorrect because it suggests a higher degree of both x and y than what exists in any of the terms. The actual common factors must be less than or equal to the degrees in the polynomial's terms.
The greatest common factor of the polynomial 12x^4y^2 + 18x²y^4 + 24x³y is 6x²y. This factor is derived from identifying the highest coefficients and the lowest powers of the variables in each term. It is essential to factor polynomials accurately to simplify expressions and solve equations effectively.
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