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prove that the paired degree cannot be equal to
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Q. prove that the paired degree cannot be equal to
- Assume Degree of Polynomial: Let's assume that the degree of a polynomial can be equal to for some integer .
- Degree is Whole Number: Now, the degree of a polynomial is always a whole number since it's the highest power of the variable in the polynomial.
- Term with Highest Power: If we have a polynomial with a degree of , it means that there is a term in the polynomial with the variable raised to the power of .
- Degree Calculation in Standard Form: The degree of a polynomial is the sum of the exponents of the variables in a term when the term is in its standard form. For example, the degree of is .
- Paired Degree Explanation: However, when we talk about ＂paired degree,＂ we're referring to the sum of the degrees of two polynomials when multiplied together. If we have two polynomials of degrees and , the degree of their product is .
- Addition of Degrees: If we multiply two polynomials with degrees that are multiples of , their degrees would add up to a number that is also a multiple of , since .
- Odd and Even Numbers: Adding to a multiple of will never result in another multiple of , because is an even number and is odd, so will always be odd, while is always even.
- Implication of Degree: Therefore, the degree of the product of two polynomials (paired degree) cannot be of the form , because it would imply that an even number () plus an odd number () is equal to another even number, which is not possible.
