Given a polynomial p, prove A^2-B^2 = p(1)p(-1)

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In summary, for a given polynomial p, the sum of the coefficients of the even powers is represented by A, and the sum of the coefficients of the odd powers is represented by B. By expressing A and B in terms of p(1) and p(-1), it can be proven that A^2 - B^2 = p(1)p(-1).
  • #1
Robb
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Homework Statement


Given a polynomial p, let A be the sum of the coefficients of the even powers, and let B be the sum of the coefficients of the odd powers. Prove that A^2 - B^2 = p(1)p(-1).

Homework Equations

The Attempt at a Solution


See attached. Can someone please look at my work to see if it is correct. Best!
 

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  • #2
Your solution is correct but limited. Can you show this to be true for an arbitrary number of terms ##N## without going through all the tediousness of your method? Generous hint: Can you express A and B in terms of P(1) and P(-1)?
 
  • #3
Well that was a lot easier! Thanks!
 

1. What is a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

2. What does A^2-B^2 mean in this context?

In this context, A^2-B^2 represents the difference of two squares, where A and B are the values of the polynomial at specific points.

3. What does it mean to prove A^2-B^2 = p(1)p(-1)?

To prove A^2-B^2 = p(1)p(-1) means to show that the difference of two squares, A^2-B^2, is equal to the product of the polynomial evaluated at 1 and -1, p(1)p(-1).

4. How can one prove A^2-B^2 = p(1)p(-1)?

To prove A^2-B^2 = p(1)p(-1), one can use algebraic manipulation and substitution to show that the two expressions are equivalent.

5. What is the significance of proving A^2-B^2 = p(1)p(-1)?

Proving A^2-B^2 = p(1)p(-1) can help to establish the relationship between the difference of two squares and the polynomial evaluated at specific points. It can also be used to solve problems and make predictions in various mathematical and scientific fields.

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