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Polynomials intro | Mathematics II | High School Math | Khan Academy
Khan Academy
Overview
This video introduces the concept of polynomials, breaking down the word "polynomial" into its Greek and Latin roots to explain its meaning: "many terms." It clarifies what constitutes a polynomial, emphasizing that terms must have coefficients and variables raised to nonnegative integer powers. The video also defines related terms like monomial, binomial, trinomial, degree of a term, degree of a polynomial, leading term, and leading coefficient, and demonstrates how to write polynomials in standard form.
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Chapters
- A polynomial is a mathematical expression consisting of a sum of terms.
- The word "polynomial" breaks down into "poly" (many) and "nomial" (name/term), meaning "many terms."
- Each term in a polynomial is a coefficient multiplied by a variable raised to a nonnegative integer power.
- Constant numbers (like 9) can be considered polynomials, as they can be written as the number times a variable to the power of zero (e.g., 9x^0).
Understanding the fundamental definition of a polynomial is crucial for recognizing and working with these expressions in various mathematical contexts.
10x^7 - 9x^2 + 15x^3 + 9 is a polynomial with four terms.
- Expressions with variables raised to negative powers are not polynomials (e.g., 10x^-7).
- Expressions with variables raised to fractional powers are not polynomials (e.g., 9a^(1/2)).
- Expressions where the exponent itself is a variable are not polynomials (e.g., 9a^a).
Knowing what disqualifies an expression from being a polynomial helps solidify the understanding of the defining characteristics of polynomials.
10x^-7 - 9x^2 + 15x^3 + 9 is not a polynomial because of the x^-7 term.
- A term is composed of a coefficient and a variable raised to a power.
- The coefficient is the number that multiplies the variable.
- The exponent of the variable must be a nonnegative integer (0, 1, 2, 3, ...).
Identifying the coefficient and the exponent in each term is essential for classifying polynomials and understanding their properties.
In the term 10x^7, 10 is the coefficient and 7 is the nonnegative integer exponent.
- A monomial is a polynomial with exactly one term (e.g., 6x^0, 10z^15).
- A binomial is a polynomial with exactly two terms (e.g., 9a^2 - 5, 3y^3 + 5y).
- A trinomial is a polynomial with exactly three terms (e.g., 10x^7 - 9x^2 + 15x^3).
These specific names (monomial, binomial, trinomial) provide a shorthand for describing polynomials based on their structure, aiding in communication and analysis.
The expression 7y^2 - 3y + pi is a trinomial because it has three terms.
- The degree of a term is the exponent of its variable.
- The degree of a polynomial is the highest degree among all its terms.
- Standard form means writing the terms of a polynomial in descending order of their degrees.
- The leading term is the first term when a polynomial is written in standard form.
- The leading coefficient is the coefficient of the leading term.
Understanding degree and standard form is crucial for comparing polynomials, performing operations, and analyzing their behavior.
The polynomial 10x^7 + 15x^3 - 9x^2 + 9 is in standard form, with a leading term of 10x^7 and a leading coefficient of 10.
Key takeaways
- Polynomials are built from terms, where each term involves a coefficient and a variable raised to a nonnegative integer power.
- The key restrictions for an expression to be a polynomial are that exponents must be nonnegative integers, not negative numbers, fractions, or variables.
- The number of terms in a polynomial gives it specific names: monomial (1 term), binomial (2 terms), and trinomial (3 terms).
- The degree of a polynomial is determined by the highest exponent found among its terms.
- Writing polynomials in standard form (highest degree first) makes it easier to identify the degree, leading term, and leading coefficient.
Key terms
PolynomialTermCoefficientVariableExponentNonnegative integerMonomialBinomialTrinomialDegree of a termDegree of a polynomialStandard formLeading termLeading coefficient
Test your understanding
- What are the two main conditions that a variable's exponent must satisfy for an expression to be considered a polynomial?
- How does the definition of a "term" in a polynomial relate to the concept of "many names" derived from the word "polynomial"?
- Why is it important to write polynomials in standard form?
- How can you determine the degree of a polynomial, and what is the significance of the leading term and leading coefficient?