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I am doing a course in algebra, in the course professor is teaching " The degree of the sum is the larger of the two starting degrees" which means if we add two polynomial having the same degree then after doing the sum the total degree will add up. I am finding the flaw in the above quoted statement. How come the degree of the sum will be larger it has to be same.

If we add $x^2 + x^2$ the sum will be $2x^2$, degree remains same.

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    $\begingroup$ Maybe the quotation refers to multiplication? 'cause addition does not have such property. $\endgroup$
    – xbh
    Aug 25, 2018 at 7:06
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    $\begingroup$ The degree of the sum is the larger of the two starting degrees Please quote exactly what the professor said, and in what context. What you wrote is obviously false, for example $\,(x)+(-x)=0\,$. $\endgroup$
    – dxiv
    Aug 25, 2018 at 7:06
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    $\begingroup$ It is not "larger than the two staring degrees", it is the "larger of the two starting degrees" $\endgroup$ Aug 25, 2018 at 7:07
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    $\begingroup$ As you're quoting it, his statement talks about "the larger of the two starting degrees", in your example the degree of both addends are $2$, the larger being $2$ which is also the degree of the sum. The statement still isn't correct though, $x^2+x$ and $-x^2+1$ both have degree $2$, but the sum has degree $1$ which is smaller than $2$. $\endgroup$ Aug 25, 2018 at 7:12
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    $\begingroup$ The word "larger" may imply here that the degrees are unequal, so that there is no rule implied for equal degrees. The formulation is ambiguous. If the degrees are equal, say both $n$, then the degree of the sum is $\le n$. $\endgroup$ Aug 25, 2018 at 7:26

2 Answers 2

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If it's not just a poor and unfortunate translation (your profile doesn't say where you come from, so I don't know if that's a possibility), it appears as if you have misunderstood your professor.

There's a difference between "larger than" and "larger of".

The degree of the sum of two polynomials will always be equal to or smaller than the larger of the degrees of the addends, in most cases, e.g. if the two addends does not have the same degree, it will be equal. There are examples in the comments and in gimusi's answer of it becoming smaller, the thing being that the two terms of highest degree cancel each other because their coefficients add to zero.

As in got mentioned in a comment: If we're dealing with multiplication, the two terms of the highest degree (let's say $ax^b$ and $cx^d$) get multiplied and the result has degree $b+d$ and the coefficient is $ac$ which is not zero so the degree of the result is always $b+d$, i.e. the sum of the degrees.

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  • $\begingroup$ +1 Spot on. In your last paragraph, you assme $ac\neq0$? $\endgroup$
    – Servaes
    Aug 25, 2018 at 15:56
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    $\begingroup$ In order not to confuse the OP, I chose not to specify that I was assuming that coefficients came from a ring without zero divisors (like $\mathbb R$), in which case $ac=0$ would imply that either $a$ or $c$ was $0$ contradicting that they came from the highest order term of the two polynomials. $\endgroup$ Aug 25, 2018 at 16:13
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No we have that the degree of the sum is less than or equal to the maximum of the two starting degrees, let us consider as an example

$$(x^2+2x+1)+(-x^2+3)=2x+4$$

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    $\begingroup$ The statement quoted by the OP does, however, hold whenever the two polynomials being summed are of different degree. $\endgroup$ Aug 25, 2018 at 10:36

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