Introduction to Polynomials

Professor Dave Explains
16 Sept 201705:12
EducationalLearning
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TLDRThis script introduces polynomials, which are algebraic expressions containing variables raised to positive whole number exponents. It covers the structure and conventions of writing polynomials, including naming monomials, binomials, trinomials, and expressions by their degree. It explains how to evaluate polynomials by plugging in values. Finally, it notes that while evaluating polynomials is straightforward, solving polynomial equations can become extremely complicated at higher degrees, requiring intricate techniques that mathematicians devote lifetimes to studying.

Takeaways
  • 😀 Polynomials are expressions with variables raised to positive whole number exponents
  • 😎 The order of terms in a polynomial goes from highest to lowest exponent
  • 📚 The degree of a polynomial is the exponent on its leading term
  • 🤓 Quadratic polynomials have degree 2, cubic have degree 3
  • 🧐 A polynomial can be missing terms, which are treated as 0
  • 👍 Evaluating polynomials involves plugging in the value for the variable
  • 😕 Solving polynomial equations can get extremely complicated
  • 🤯 Brilliant mathematicians work on solving complex polynomials their whole lives
  • 😀 We will start by focusing on the basics of working with polynomials
  • 🤠 Put some reverence for algebra's complexity in your pocket as we continue
Q & A

  • What are polynomials?

    -Polynomials are algebraic expressions that contain an unknown variable raised to various positive whole number exponents, such as x^2 + 4x + 10.

  • What are the key components that make up a polynomial?

    -The key components are: a variable (x), exponents representing the powers the variable is raised to, coefficients multiplying each term, constants or numbers with no variable.

  • How are the terms in a polynomial typically ordered?

    -By convention, polynomial terms are listed in decreasing order of their exponents.

  • What is the degree of a polynomial?

    -The degree of a polynomial refers to the highest power or exponent present for the variable term. For example, x^3 + 2x + 5 has degree 3.

  • What are some special names given to low-degree polynomials?

    -A monomial has one term, a binomial has two terms, and a trinomial has three terms. Higher degree polynomials do not have special names.

  • Do polynomials have to contain all possible terms?

    -No, polynomials can have missing terms, which are treated as if they have a coefficient of 0.

  • How do you evaluate a polynomial for a given value of x?

    -To evaluate, substitute the given x-value for x in the polynomial expression and compute using normal algebraic order of operations.

  • Why can polynomial equations be hard to solve for x?

    -The presence of multiple x terms with exponents means we cannot use typical algebraic solution methods to isolate x on one side.

  • What are some ways to solve polynomial equations?

    -Factoring, completing the square, using the quadratic formula, synthetic division, and numerical approximation methods can be used to solve certain polynomial equations.

  • How complex can polynomial solutions get?

    -Very complex! As the degree increases, the solutions can become extremely intricate, requiring advanced techniques that mathematicians research and specialize in.

Outlines
00:00
😃 Introducing Polynomials

This paragraph introduces polynomials, explaining that they are algebraic expressions containing an unknown variable raised to positive whole number exponents. It gives examples of monomials, binomials, and trinomials based on the number of terms. It also notes conventions like arranging terms by decreasing exponents and defining the degree of a polynomial.

😄 Evaluating and Solving Polynomials

This paragraph contrasts evaluating and solving polynomials. Evaluating for a given value of the variable is straightforward. But solving to find the variable's values that satisfy the equation can be extremely complicated, requiring advanced techniques. It notes reverence for the deep complexity algebra enables while setting up forthcoming coverage of the basics.

Mindmap
Keywords
💡polynomial
A polynomial is an algebraic expression containing one or more terms with variables raised to positive integer exponents. Polynomials are a key concept in algebra and are the main focus of this video. Examples of polynomials from the script include 'two X squared plus four X plus ten' and 'three X cubed plus five X squared minus two X plus seven'.
💡degree
The degree of a polynomial refers to the highest exponent or power that the variable is raised to within the polynomial. For example, in the polynomial 'two X squared plus four X plus ten', the highest power of X is 2, so this is called a second degree or quadratic polynomial. Knowing the degree gives information about the complexity of the polynomial.
💡monomial
A monomial is a simple polynomial with only one term. For example, 'two X' would be considered a monomial. Identifying whether a polynomial expression is a monomial, binomial, trinomial, or has even more terms is useful for classifying and analyzing it.
💡binomial
A binomial is a polynomial with two terms, like 'two X minus one'. Recognizing an expression as a binomial reveals that it has a specific form with two distinct parts, which can help when working with the polynomial.
💡trinomial
A trinomial is a polynomial with three terms, such as 'X squared plus two X minus one'. Realizing an expression is a trinomial tells us to expect three parts that will need to be handled in any manipulation or solving of the polynomial.
💡evaluate
To evaluate a polynomial means to substitute a specific number for the variable(s) and simplify the expression to get a numerical result. The video shows how to evaluate 'two X squared plus four X plus ten' when X is -1, by plugging in -1 for X throughout.
💡solve
Solving a polynomial means finding the values for the variable(s) that make the polynomial equal to zero (or some other value). This is more complex than evaluating and often requires advanced techniques, which the video says will be covered later.
💡exponent
An exponent indicates the power to which a variable or number is raised in a polynomial term. For example, the 'X squared' term has an exponent of 2 on the X variable. Exponents are useful for classifying terms and analyzing a polynomial.
💡coefficient
The numerical factor multiplied by a variable or variables in a term is called the coefficient. For example, in 'two X squared', the number 2 is the coefficient. Coefficients help determine the contribution of each term to the polynomial.
💡term
A term in a polynomial is a number, variable, or product of a number and one or more variables that are added or subtracted. For example, 'two X', 'four X', and 'ten' are all terms in the polynomial 'two X squared plus four X plus ten'. Analyzing and manipulating the individual terms is key to working with polynomials.
Highlights

Polynomials are expressions that contain an unknown raised to various positive whole number exponents.

Polynomials follow a general form with decreasing exponents and a constant term.

Polynomials with one term are monomials, two terms are binomials, and three terms are trinomials.

The highest power present in a polynomial is called its degree.

We can evaluate a polynomial by plugging in a value for the variable.

Solving polynomial equations can be extremely complicated compared to other algebraic equations.

Even brilliant mathematicians spend lifetimes trying to figure out ways to solve complex polynomials.

It's good to show some reverence for the deep complexity algebra is capable of.

There are a variety of ways to solve polynomial equations that we will learn.

If there is just one term, like two X, it’s a monomial.

We typically only interact with expressions that go up to a power of three or four.

Polynomials don’t have to have all the possible terms.

Polynomials get much more complex than this, their solutions become so intricate.

Strategies we have learned thus far will not work to solve polynomials.

Put reverence for algebra's complexity in your pocket as we've got more polynomial work.

Transcripts

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