Algebra 83 - Polynomials

TL;DR

Polynomial functions are powerful mathematical tools that model real world problems and can be used in physics, economics, and social science.

Transcript

Hello. I'm Professor Von Schmohawk and welcome to Why U. So far in our lectures, we have studied several types of functions of a real variable. In particular, we saw that linear functions have graphs that are lines and that quadratic functions have graphs that are parabolas. Both of these functions are examples of a broader class of functions calle... Read More

Key Insights

• 🌍 Polynomial functions can mathematically model real world problems in physics, economics, and social science.
• 🏛️ Monomials are the building blocks of polynomial functions, containing variables, coefficients, and non-negative integer exponents.
• 🍉 The terms in a polynomial function are ordered so that the exponents decrease from left to right, with the leading term determining the degree of the polynomial.
• 😑 Polynomial expressions can be represented in general notation, with the coefficients and exponents organized in a specific pattern.
• ✋ Zero-degree, first-degree, and second-degree polynomial functions correspond to constant, linear, and quadratic functions respectively, with more complex and varied shapes for higher-degree polynomial functions.
• 📈 Polynomial functions can be used to analyze and interpret various phenomena, such as graphs of projectile motion, economic growth, or population growth.
• 👻 Negative or fractional exponents are not allowed in polynomial functions, as they represent division or roots respectively.

Questions & Answers

Q: What are examples of real world problems that polynomial functions can model?

Polynomial functions can be used to model problems in physics, economics, and social science. For example, they can be used to model projectile motion, economic growth, or population growth.

Q: What is the difference between a monomial, binomial, and trinomial?

A monomial is a polynomial with only one term, a binomial has two terms, and a trinomial has three terms. The names correspond to the number of terms in the polynomial.

Q: Can the exponents in a polynomial be negative or fractional?

No, the exponents in a polynomial must be non-negative integers. Negative exponents represent division (such as 1/x) and fractional exponents represent roots (such as square root or cube root).

Q: How can a polynomial expression be represented in general notation?

In general notation, the leading term is written as "a sub n" times "x to the nth power", followed by terms with decreasing exponents. The constants, represented as "a sub n", can be positive, negative, or zero.

Summary & Key Takeaways

• Polynomial functions are a type of function that can model various real world problems.

• The terms in a polynomial function are called monomials, which are a single term containing variables and coefficients.

• Polynomials are ordered so that the exponents decrease from left to right, and the leading term determines the degree of the polynomial.