A polynomial is an algebraic expression that involves sums and/or differences of terms. Each term in a polynomial consists of a coefficient (a number) and a variable raised to an exponent (a power). The exponents are non-negative integers (whole numbers like 0, 1, 2, 3, etc.).

The general form of a polynomial is:

axⁿ + bx^n-1 + cx^n-2 + ... + k

In this expression:

• a, b, c, ... are the coefficients of the terms.
• n, n-1, ... are the exponents of the variables.
• k is a constant term (a term with no variable).

What Makes an Expression a Polynomial?

A polynomial must satisfy the following:

• The variable (typically x) can only have whole-number exponents.
• The exponents cannot be negative or fractional.
• The terms of the polynomial are added or subtracted together.

Types of Polynomials

Polynomials are classified based on the degree (the highest exponent) of the variable:

• Constant Polynomial: A polynomial with no variable, just a constant (e.g., 5).
• Linear Polynomial: A polynomial of degree 1 (e.g., 3x + 2).
• Quadratic Polynomial: A polynomial of degree 2 (e.g., x² - 5x + 6).
• Cubic Polynomial: A polynomial of degree 3 (e.g., 2x³ + x² - 3x + 1).
• Quartic Polynomial: A polynomial of degree 4 (e.g., x⁴ - 2x³ + x - 5).

Example 1: Simplifying a Polynomial

Simplify the polynomial: 3x² + 5x - 2 + 4x² - 3x + 1
Step 1: Combine like terms (terms with the same variable and exponent):

• 3x² + 4x² = 7x²
• 5x - 3x = 2x
• -2 + 1 = -1

Step 2: The simplified polynomial is: 7x² + 2x - 1

Example 2: Evaluating a Polynomial

Evaluate the polynomial 2x² - 3x + 5 when x = 3.

• Step 1: Substitute x = 3 into the polynomial.
• 2(3)² - 3(3) + 5
• Step 2: Simplify.
• 2(9) - 3(3) + 5
• 18 - 9 + 5
• 18 - 9 = 9, and 9 + 5 = 14

So, when x = 3, the value of the polynomial is 14.

Summary:

A polynomial is an expression made up of terms with non-negative integer exponents. Polynomials can be classified based on their degree, such as linear, quadratic, or cubic. To work with polynomials, you can simplify them by combining like terms, or you can evaluate them by substituting a value for the variable.

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