Prove Product of Polynomials: No Odd Degree Terms

In summary, proving that the product of polynomials has no odd degree terms means showing that the resulting polynomial after multiplying two or more polynomials does not contain any terms with an odd degree. This process involves using the distributive property, combining like terms, and checking the degree of each term in the resulting polynomial. It is important because it helps simplify the polynomial and understand its structure. While there can be exceptions, this concept relates to the fundamental theorem of algebra as it helps determine the number of real and complex roots of the polynomial.
  • #1
anemone
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Prove that in the following product

$P=(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$

after multiplying and collecting like terms, there does not appear a term in $x$ of odd degree.
 
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  • #2
I call it P(x) a polynomial

$P(x) =(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$

So $(1-x^2)P(x) =(1-x)(1+x) P(x) = (1+x) (1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1-x) (1+x+x^2+x^3+\cdots+x^{99}+x^{100})$
OR
$(1-x^2)P(x) =(1+x^{101})(1-x^{101})=(1-x^{202})$

OR
$P(x) =\frac{1-x^{202}}{1-x^2} = (1+x^2 + x^4+x^6+\cdots+x^{198}+x^{200})$

So no x term with odd degree
 

FAQ: Prove Product of Polynomials: No Odd Degree Terms

How do you prove that the product of two polynomials has no odd degree terms?

To prove that the product of two polynomials has no odd degree terms, you can use the distributive property and the fact that multiplying an even number by an odd number always results in an even number. This means that when multiplying two polynomials, any odd degree terms will always be multiplied by an even degree term, resulting in an even degree term. Therefore, there can be no odd degree terms in the product of two polynomials.

Can you provide an example of a polynomial product with no odd degree terms?

Yes, an example of a polynomial product with no odd degree terms is (x^2 + 3x + 2)(x^4 + 2x^2 + 1). When expanded using the distributive property, the product becomes x^6 + 5x^4 + 7x^2 + 2, which only contains even degree terms.

Is it possible for a product of polynomials to have no odd degree terms if the individual polynomials both have odd degree terms?

No, it is not possible for a product of polynomials to have no odd degree terms if the individual polynomials both have odd degree terms. This is because when multiplying two polynomials with odd degree terms, the resulting product will always have at least one odd degree term, as the exponents of the terms will add up to an odd number.

How does proving that the product of polynomials has no odd degree terms relate to the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra states that every polynomial of degree n has n complex roots. In order for a polynomial to have complex roots, it must have at least one odd degree term. Therefore, if the product of two polynomials has no odd degree terms, it cannot have any complex roots, and therefore does not satisfy the Fundamental Theorem of Algebra.

Can you use the principle of mathematical induction to prove that the product of polynomials has no odd degree terms?

Yes, you can use the principle of mathematical induction to prove that the product of polynomials has no odd degree terms. The base case would be a polynomial product with only one term, which would have no odd degree terms. Then, using the inductive hypothesis that the product of two polynomials with no odd degree terms also has no odd degree terms, you can show that the product of n+1 polynomials also has no odd degree terms.

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