When dividing a third degree polynomial (let's denote it as \( P(x) \)) by a second degree polynomial (let's denote it as \( D(x) \)), we can analyze the possible outcomes.
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Degree of the Quotient: The degree of the quotient will be \( \text{deg}(P) - \text{deg}(D) = 3 - 2 = 1 \). Thus, the quotient will be a first degree polynomial or a linear polynomial.
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Remainder: By the polynomial division algorithm, when dividing polynomials, there is always a remainder, which can be of lower degree than the divisor (hence in this case it would be of degree less than 2).
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Fraction in the Quotient: The quotient can be expressed in the form of a polynomial plus a remainder divided by the divisor. Whether there is a fraction in the quotient depends on the specific polynomials involved and their coefficients.
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Zero Term in the Quotient: A zero term can occur, but it is not guaranteed. The linear polynomial resulting from the division might not have a zero term if both polynomials do not result in such a fraction.
Based on this analysis, the statement that is always true when dividing a third degree polynomial by a second degree polynomial is:
C. the quotient is a first degree polynomial.
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