Polynomial

Definition

A polynomial is a mathematical expression that consists of variables raised to whole number powers, multiplied by coefficients, and added or subtracted together. It can be represented in the form of $$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$, where each term is made up of a coefficient and a variable raised to a non-negative integer exponent. Polynomials play a crucial role in various mathematical concepts, including power functions, which are specific types of polynomials.

5 Must Know Facts For Your Next Test

1. Polynomials can have multiple terms, and they are categorized based on their degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.
2. The general form of a polynomial includes coefficients that can be any real number and exponents that are non-negative integers.
3. Addition, subtraction, and multiplication can be performed on polynomials, making them very versatile in mathematical operations.
4. Power functions are considered specific types of polynomials where the exponent is a single non-negative integer, such as $$f(x) = kx^n$$.
5. When graphing polynomials, their shapes and behavior are influenced by their degree and leading coefficient, impacting the number of turns and end behavior.

Review Questions

• How do polynomials relate to power functions, and what defines a power function?
  • Polynomials include power functions as specific cases where the variable is raised to an integer exponent. A power function generally takes the form $$f(x) = kx^n$$, where 'k' is a coefficient and 'n' is a non-negative integer. All power functions are polynomials; however, not all polynomials can be classified as power functions since they may contain multiple terms with different exponents. Understanding this relationship helps in recognizing how power functions can be viewed as building blocks within the broader category of polynomials.
• Evaluate how the degree of a polynomial affects its graph and identify key characteristics based on its degree.
  • The degree of a polynomial significantly influences its graph. For example, linear polynomials (degree 1) produce straight lines, while quadratic polynomials (degree 2) create parabolas. Higher degree polynomials can have multiple turning points and will display varying end behaviors based on the leading coefficient. Generally, even-degree polynomials will have both ends pointing in the same direction, while odd-degree polynomials will point in opposite directions. Recognizing these patterns is essential for predicting graph behaviors.
• Create a polynomial from a given scenario involving area calculation and explain how it exemplifies polynomial properties.
  • Suppose you have a rectangular garden where the length is represented as $$x + 3$$ meters and the width as $$x$$ meters. The area of this garden can be expressed as a polynomial: $$A(x) = (x + 3)x = x^2 + 3x$$. This polynomial demonstrates key properties such as having two terms (making it quadratic) and coefficients of 1 for $$x^2$$ and 3 for $$x$$. By identifying the length and width in this context, we see how polynomials model real-world scenarios effectively.

Related terms

Monomial: A monomial is a polynomial with only one term, typically represented as $$ax^n$$ where 'a' is a coefficient and 'n' is a non-negative integer.

Degree: The degree of a polynomial is the highest exponent of the variable in the expression, which determines the polynomial's behavior and characteristics.

Coefficient: A coefficient is a numerical factor that multiplies the variable(s) in a polynomial term, indicating how many times that term contributes to the overall polynomial.