Polynomial Inequalities

Front

What is a polynomial inequality?

Back

A polynomial inequality is an inequality that involves a polynomial expression, typically in the form of f(x) > 0, f(x) < 0, f(x) ≥ 0, or f(x) ≤ 0.

Front

How do you determine where a polynomial is positive or negative?

Back

To determine where a polynomial is positive or negative, find the roots of the polynomial, divide the number line into intervals based on these roots, and test a point from each interval in the polynomial.

Front

What does the notation U represent in interval notation?

Back

The notation U represents the union of two sets, indicating that the solution includes all values from both sets.

Front

What is the significance of the endpoints in interval notation?

Back

Endpoints indicate whether the values at those points are included in the interval (closed interval) or not (open interval). For example, [a, b] includes a and b, while (a, b) does not.

Front

Solve the inequality: x^2 - 4 > 0

Back

The solution is (-oo, -2) U (2, +oo).

Front

What is the first step in solving polynomial inequalities?

Back

The first step is to set the polynomial expression equal to zero to find the critical points.

Front

How do you test intervals for polynomial inequalities?

Back

Choose a test point from each interval created by the critical points and substitute it into the polynomial to determine if the polynomial is positive or negative in that interval.

Front

What is the difference between strict and non-strict inequalities?

Back

Strict inequalities (>, <) do not include the boundary points, while non-strict inequalities (≥, ≤) do include the boundary points.

Front

Solve the inequality: x^4 + 35x^2 > 36

Back

The solution is (-oo, -1) U (1, +oo).

Front

What is a critical point in the context of polynomial inequalities?

Back

A critical point is a value of x where the polynomial equals zero or is undefined, which helps to determine the intervals for testing.