To compare the two values for a polynomial, we need to define them clearly:
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Greatest exponent of a term with a single variable: This is simply the highest exponent of any term in a polynomial where each term consists of a single variable raised to a power. For example, in the polynomial \( p(x) = 4x^3 + 2x^5 - x + 7 \), the greatest exponent is 5.
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Greatest sum of the exponents of a term with multiple variables: This relates to terms that involve more than one variable. For a term like \( ax^m y^n \), the sum of the exponents is \( m + n \). To find the greatest sum, you would look at all the terms of the polynomial and find the maximum value of \( m + n \) for those terms. For example, in the polynomial \( p(x, y) = x^2y^3 + x^3y^2 + xy + 1 \), the term \( x^2y^3 \) has a sum of exponents \( 2 + 3 = 5 \), while the term \( x^3y^2 \) has a sum of \( 3 + 2 = 5 \).
To summarize:
- Take the greatest exponent from any term with a single variable.
- Sum the exponents of each term with multiple variables and take the greatest sum.
After you calculate both values, you can determine which is greater. Depending on the specific polynomial, one value might be greater than the other, or they could be equal.
If you're comparing the two values, you will need to analyze a specific polynomial or set of polynomials to reach a conclusion.