Symmetric and Power Sum Polynomials

The video explores a mathematical problem involving the sum of powers of variables x, y, and z. It introduces the problem, discusses its significance, and provides a systematic approach to solving it using symmetric polynomials and Newton-Girar identities. The video also highlights the importance of understanding these concepts in higher-level mathematics and concludes with a challenge for viewers.

What are the initial conditions given in the problem?

• The sum of x, y, and z is 0, the sum of their squares is 1, and the sum of their cubes is 2.

• The sum of x, y, and z is 1, the sum of their squares is 2, and the sum of their cubes is 3.

• The sum of x, y, and z is 2, the sum of their squares is 3, and the sum of their cubes is 4.

• The sum of x, y, and z is 3, the sum of their squares is 4, and the sum of their cubes is 5.

What is a symmetric polynomial?

• A polynomial that only changes for the identity permutation.

• A polynomial that remains unchanged for any permutation of the variable subscripts.

• A polynomial that is always equal to zero.

• A polynomial that changes for any permutation of the variable subscripts.

Which of the following is an example of a symmetric polynomial?

• x1 / x2

• x1 * x2

• x1 - x2

• x1 + x2

What is the relationship between power sum polynomials and elementary symmetric polynomials?

• Power sum polynomials are derived from elementary symmetric polynomials.

• They are unrelated.

• They are the same.

• Elementary symmetric polynomials are derived from power sum polynomials.

What is the kth degree power sum polynomial for n variables?

• The quotient of the variables to the kth power.

• The difference of the variables to the kth power.

• The product of the variables to the kth power.

• The sum of the variables to the kth power.

Who discovered the identities that relate elementary symmetric polynomials to power sum polynomials?

• Both Newton and Girar

• Neither Newton nor Girar

• Albert Girar

• Isaac Newton

What is the value of e1 in the given problem?

• 3

• 2

• 1

• 0

What is the value of e2 in the given problem?

• 1

• -1

• 0.5

• -0.5

What is the value of p4 in the given problem?

• 1/6

• 0

• 6

• 25/6

What is the final value of the sum of the fifth powers?

• 6

• 5

• 8

• 7