2.6. Summary

A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = anxn + an–1xn – 1 + . . . + a2x2 + a1x + a0, where a0, a1, a2, . . ., an are constants and an ≠ 0.

A polynomial is a specific type of mathematical expression that includes variables raised to whole number powers and coefficients. The general form is represented by p(x), where x is the variable. Each coefficient (like a0, a1) corresponds to a specific power of x. For example, if we have p(x) = 2x² + 3x + 5, the coefficients are 2, 3, and 5 corresponding to x², x, and the constant term respectively. The highest power of x in a polynomial determines the polynomial's degree, which must have a leading coefficient that is not zero (an ≠ 0).

Think of a polynomial like a recipe for a cake. Each ingredient (coefficient) contributes to the final flavor (resulting polynomial) depending on its amount and combination with other ingredients (terms). Just as a recipe requires certain conditions (like correct ingredient amounts), a polynomial must be structured according to specific rules (such as having a leading non-zero coefficient).

1. A polynomial of one term is called a monomial.
2. A polynomial of two terms is called a binomial.
3. A polynomial of three terms is called a trinomial.

Polynomials can be classified based on the number of terms they contain. A monomial has just one term, such as 4x or 7. A binomial has two terms, like 3x + 5. A trinomial has three terms, for instance, x² + 2x + 1. This classification helps in simplifying and factoring expressions and solving equations.

Think of terms in a polynomial as different types of fruits in a fruit salad. A monomial is like a salad containing only one kind of fruit, a binomial contains two types of fruits (e.g., apples and bananas), and a trinomial has three types (like apples, bananas, and oranges). Just as different combinations create different tastes in the salad, different combinations of polynomial terms create unique expressions.

5. A polynomial of degree one is called a linear polynomial.
6. A polynomial of degree two is called a quadratic polynomial.
7. A polynomial of degree three is called a cubic polynomial.

The degree of a polynomial is the highest exponent of the variable in the expression. A linear polynomial (degree one) takes the form mx + b, which graphs as a straight line. A quadratic polynomial (degree two), represented by ax² + bx + c, creates a parabolic shape when graphed. A cubic polynomial (degree three) has the form ax³ + bx² + cx + d and draws more complex curves. Understanding the degree helps predict how the polynomial behaves.

Visualize climbing a hill. A linear polynomial is like a gentle slope; you simply go up or down. A quadratic is like a grassy hill with a peak, where you go up to a point and then down. A cubic polynomial resembles a hilly roller coaster with ups and downs, making for a more exhilarating ride.