Solve each polynomial inequality in Exercises 1-42 and graph the solution set on a real number l

Solve each polynomial inequality in Exercises 1-42 and graph...

Key Concepts

Graphing on the Real Number Line

Graphing the solution set on the real number line provides a visual representation of where the inequality holds true. This involves marking the critical points and shading the intervals that satisfy the inequality condition, offering a clear and intuitive understanding of the solution.

Interval Notation

Interval notation is a standardized mathematical notation used to represent subsets of the real number line succinctly. It is particularly useful for expressing the solution sets of inequalities, indicating which endpoints are included or excluded based on whether the inequality is strict or non-strict.

Sign Chart Analysis

A sign chart is used to determine the sign (positive or negative) of a polynomial on the intervals determined by its critical points. By selecting sample points within these intervals, one can easily test whether the polynomial satisfies the inequality condition, allowing for the correct identification of solution intervals.

Critical Points

Critical points are the real roots of the polynomial where the expression equals zero. They are crucial because they divide the number line into intervals over which the polynomial's sign is consistent. Identifying these points helps in establishing the boundaries for the inequality analysis.

Factoring Polynomials

Factoring is a key technique to simplify a polynomial expression by breaking it down into simpler polynomial factors, which reveals its roots. These factors can often be found using techniques such as synthetic division, factoring by grouping, or recognizing familiar patterns, and are essential in determining the critical points for the inequality.

Polynomial Inequality

A polynomial inequality is an expression involving a polynomial set in relation to zero by an inequality sign. Solving such inequalities involves finding the regions where the polynomial takes positive or negative values. The process generally requires finding the roots of the polynomial and then analyzing how these roots divide the number line into intervals with consistent signs.

Transcript

00:01 Okay, for this problem, we've got a third -degree polynomial on the left and zero on the right.

00:06 So to start, we're going to want to keep track of all of the positive and negative factors of our constant term.

00:13 And unfortunately, we're going to have a lot here.

00:15 We're going to have one, three, nine, and 27.

00:24 So i guess not too many, but we have plus and minus, so we'll have eight total that we would have to think about.

00:31 And what we're going to do is we're going to just pick one of them to start with.

00:36 So we'll start with positive one maybe.

00:39 And we're going to write out the constant or the coefficients of our polynomial.

00:45 Like so, 1, negative 3, negative 9, 27.

00:51 And we're going to do the process by which we can decide if this is a root of our polynomial.

00:58 So we start by just bringing this first term straight down, multiplying here.

01:03 1 times 1 gives us 1.

01:06 Adding these together, we get negative 2.

01:08 Multiplying these gives us negative 2.

01:11 Adding these together gives us negative 11.

01:13 Multiplying gives us negative 11.

01:16 Adding gives us 16.

01:17 The important thing here is that it is not 0, so negative 1 was not a root of our polynomial.

01:25 So what we're going to do? so we're going to erase these numbers, and we're going to try the next thing.

01:32 If negative 1 didn't work, or if positive, one didn't work, maybe positive three will work.

01:38 So we're going to bring this one straight down again.

01:42 And up here, this time we'll have a three.

01:44 So we're going to take three times one, gives us a three...

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