Why do you think the following statements are true or false? Demonstrate whether each
of them is true or false for polynomials over a field:
1. The product of monic polynomials is monic
2. The product of polynomials of degrees m and n has degree m + n
The sum of polynomials of degree m and n has degree max[m, n] where m ≠ n
Question 1:
True – In order for a polynomial to be monic it must have a leading
coefficient of 1. Therefore, I will use plain English and an example to show the proof.
In
order to multiply two polynomials add their exponents together and then multiply their
coefficients.
This will produce their product.
For example, the product of 2x
2
* 2x
3
=
4x
6
.
This is done by taking the exponents 2 and 3 and adding them together to make 6.
Next multiply the coefficients 2 * 2 which produces 4.
Thus, the product is 4x
6
, while if
you multiply two monics such as x
2
*x
2
, you would get x4; thus, if the leading coefficients
are 1 as required to be monic then the product of the coefficients of 1*1 will always be
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 Spring '08
 Philip
 Algebra, Polynomials, Multiplication, Coefficient

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