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When we discuss polynomial functions, we're talking about expressions involving variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. A polynomial function can look as simple as a constant, like 5, or it can be a more complex combination like
\( 3x^4 - 2x^3 + 7x - 5 \).
The highest power of the variable in the polynomial indicates its degree, and this degree tells us much about the behavior of the function, especially its graph. For instance, the polynomial
\( x^2 - 4x + 4 \)
is a second-degree polynomial, also known as a quadratic, and its graph is a parabola. When working with polynomial inequalities, we are usually interested in the intervals where the polynomial is positive or negative, which are crucial in defining solutions to inequalities.

Inequality symbols are the shorthand for the mathematical language that defines relationships involving quantities that aren’t equal. The most common inequality symbols are:

• Less than (<)
• Greater than (>)
• Less than or equal to (≤)
• Greater than or equal to (≥)

\( < \) and \( > \) are strict inequalities, meaning that one side is definitively greater or less than the other. \( ≤ \) and \( ≥ \) are non-strict inequalities, where the quantities might be equal or, respectively, less than or greater than the other. These symbols are fundamental to understanding polynomial inequalities, as they tell us on which side of the graph our solutions lie and whether or not to include the endpoints in our interval solution.

Quadratic polynomials are second-degree polynomials and typically take the general form
\( ax^2 + bx + c \),
where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The graph of a quadratic polynomial is always a parabola, which can open upwards or downwards depending on the sign of \( a \). The quadratic polynomial plays a central role in polynomial inequalities like
\( x^2 + 2x - 3 < 0 \).
Solving this inequality would involve finding the values of \( x \) for which the parabola's y-values are below the x-axis. It’s a key concept in algebra because it sets the stage for understanding more complex polynomial inequalities.