When dividing polynomials, the degree of the remainder [tex]R(x)[/tex] must be what in relation to the degree of the dividend?
A. Less than
B. Equal to
C. None of these
D. Greater than
Answer (2)
The degree of the remainder in polynomial division is always less than the degree of the divisor.
There is no fixed relationship between the degree of the remainder and the degree of the dividend.
The correct answer is 'None of these'.
Therefore, the degree of the remainder R ( x ) must be None of these the degree of the dividend.
Explanation
Understanding the Problem When dividing polynomials, the degree of the remainder R ( x ) must be compared to the degree of the divisor, not the dividend. The question asks to choose from 'Less than', 'Equal to', 'None of these', 'Greater than' to describe the relationship between the degree of the remainder and the degree of the dividend.
Polynomial Division Algorithm The polynomial division algorithm states: Dividend = (Divisor * Quotient) + Remainder. The degree of the remainder must always be strictly less than the degree of the divisor.
Relationship Between Degrees The degree of the dividend is equal to the sum of the degrees of the divisor and the quotient. Therefore, the degree of the remainder must be less than the degree of the divisor.
No Fixed Relationship Since the degree of the divisor can be less than, equal to, or greater than the degree of the dividend, there is no fixed relationship between the degree of the remainder and the degree of the dividend.
Final Answer Therefore, the correct answer is 'None of these'.
Examples
Polynomial division is used in various applications, such as simplifying complex rational expressions in calculus and engineering. For example, when analyzing the stability of a control system, engineers often need to decompose a transfer function into simpler terms using polynomial division. This allows them to identify potential issues and design effective control strategies. Understanding the relationship between the degrees of the dividend, divisor, quotient, and remainder is crucial for these applications.
Approximately 2.81 × 1 0 21 electrons flow through an electric device delivering a current of 15.0 A for 30 seconds. This calculation is based on the total charge that flows and the charge of a single electron. By using the relationship between current, charge, and time, we can determine the total number of electrons.
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