Polynomial

Definition

A polynomial is a mathematical expression made up of variables, coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication. It can be represented in standard form, where the terms are ordered by descending powers of the variable. Polynomials are fundamental in algebra and are used in various applications, including modeling real-world situations and solving equations.

5 Must Know Facts For Your Next Test

1. Polynomials can have one or more terms, which can be classified as linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
2. The general form of a polynomial in one variable is given by $$P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$, where $$a_n$$ is the leading coefficient and $$n$$ is the degree.
3. Polynomials can be added, subtracted, multiplied, and divided (except by zero) while following specific rules for combining like terms.
4. A polynomial with two variables can be expressed as $$P(x,y) = a_{ij} x^i y^j$$, where the coefficients $$a_{ij}$$ correspond to different combinations of variables.
5. Polynomials play an important role in calculus, especially when finding limits, derivatives, and integrals.

Review Questions

• How do you identify the degree of a polynomial and why is it significant?
  • To identify the degree of a polynomial, look for the highest exponent on the variable within the expression. The degree is significant because it indicates the polynomial's behavior as the variable approaches infinity or negative infinity. It also helps in classifying the polynomial type (linear, quadratic, cubic, etc.) and plays a crucial role in understanding its graph and roots.
• Compare and contrast monomials with polynomials. What are their differences in terms of structure and operations?
  • Monomials are specific types of polynomials that consist of only one term, whereas polynomials can have multiple terms combined through addition or subtraction. In operations, both monomials and polynomials follow similar rules for multiplication and division. However, when adding or subtracting polynomials, you must combine like terms while this does not apply to monomials since they only contain one term.
• Evaluate how factoring polynomials can help solve equations. What techniques are commonly used in this process?
  • Factoring polynomials is an essential technique for solving equations as it allows us to express a complex polynomial in simpler forms. By setting each factor equal to zero (using the Zero Product Property), we can find solutions or roots of the original equation. Common techniques for factoring include identifying common factors, using methods such as grouping or applying special formulas like the difference of squares or perfect square trinomials to break down polynomials into simpler components.

Related terms

Monomial: A monomial is a polynomial that consists of a single term, which includes a coefficient and a variable raised to a non-negative integer exponent.

Degree of a Polynomial: The degree of a polynomial is the highest power of the variable present in the polynomial, which helps determine its behavior and properties.

Factoring: Factoring is the process of breaking down a polynomial into simpler polynomials or factors that can be multiplied together to yield the original polynomial.