EE102: Introduction to Signals & Systems, Lecture 9

TL;DR

Rational functions can be expressed in partial fraction expansion form, which facilitates computation and understanding of the function.

Transcript

you're done the pad well marching right along I guess the last time we started by just sort of reviewing what's a polynomial what's it what's a root of a polynomial what's the multiplicity of a root and this leads led us to fundamental theorem of algebra and that says that if you have a polynomial of degree n it has exactly n roots counting multipl... Read More

Key Insights

• 🎭 Polynomials can be represented in either coefficient form or factored form, both of which have advantages depending on the operation being performed.
• 💦 Complex roots of a polynomial with real coefficients come in complex conjugate pairs, making it easier to work with them.
• 😑 Rational functions can be expressed in partial fraction expansion form, which facilitates computation and understanding of the function.
• 💈 Finding the partial fraction expansion involves finding the poles and residues of the function, which can be accomplished through various methods.

Questions & Answers

Q: What is the advantage of writing a polynomial in factored form?

Writing a polynomial in factored form makes it easier to determine if one polynomial is a multiple of another. It also simplifies evaluating a polynomial at a given complex number.

Q: Why do complex roots of a polynomial with real coefficients come in complex conjugate pairs?

Complex roots of a polynomial with real coefficients come in complex conjugate pairs because if a complex number is a root of a polynomial, its conjugate is also a root. This property is not applicable to real roots.

Q: What is the significance of the fact that the coefficients of the polynomials are real?

The fact that the coefficients of the polynomials are real means that whenever a complex number is a root of a polynomial with real coefficients, its conjugate is also a root. This property allows for easier analysis and understanding of the polynomial.

Q: How can a rational function be represented in partial fraction expansion form?

To represent a rational function in partial fraction expansion form, you factor the denominator to find the poles and then find the corresponding residues using methods such as equating coefficients or the residue at a pole formula.

Summary & Key Takeaways

• A polynomial can be represented by either its coefficient form or factored form, both of which have their own advantages depending on the operation being performed.

• Complex roots of a polynomial with real coefficients come in complex conjugate pairs, making it easier to work with them.

• Rational functions, which are ratios of polynomials, can be expressed in partial fraction expansion form for easier computation and understanding.

• Finding the partial fraction expansion involves finding the poles and residues of the function, which can be accomplished through various methods.