Solve the inequality indicated using a number line and the behavior of the graph at each zero. W

step 1 Hello. Today we will be solving a polynomial inequality. This example asks us to find the values

Solve the inequality indicated using a number line and the...

Key Concepts

Interval Notation

Interval notation is a concise way of representing a set of numbers between endpoints. It is used to describe the solution set of inequalities, indicating where the inequality holds true without listing every individual point, by using brackets and parentheses to denote inclusivity or exclusivity of the endpoints.

Sign Analysis and Number Line

Sign analysis involves determining the sign (positive or negative) of the polynomial on intervals defined by its zeros. A number line is used graphically to illustrate these intervals and to analyze how the sign changes across each zero, particularly considering whether the graph crosses or touches the x-axis at those points.

Polynomial Inequality

A polynomial inequality involves finding the values of the variable that make the inequality true, where the expression is a polynomial. Solving such inequalities generally requires determining where the polynomial is positive, negative, or zero, and then identifying the intervals that satisfy the inequality.

Zeros/Roots of the Polynomial

Zeros, or roots, of a polynomial are the values of the variable for which the polynomial equals zero. They are crucial in solving inequalities because they divide the number line into intervals where the polynomial’s sign is consistent, allowing us to determine where the inequality holds.

Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler expressions. This technique is often used to find the zeros of the polynomial more easily. By breaking down the polynomial into its factors, one can set each factor to zero to solve for the roots.

Transcript

00:01 Hello.

00:02 Today we will be solving a polynomial inequality.

00:05 This example asks us to find the values of x which satisfy the inequality.

00:09 Negative 4x plus 12 is less than negative x to the 3rd plus 3x squared.

00:14 We're going to start by moving everything over to the left side and creating our function.

00:19 This will give us x to the 3x squared minus 4x plus 12 is going to be less than 0.

00:29 We're going to call this f of a third.

00:31 X and we want x of x to be less than zero okay so let's start by factoring f of x we are going to start by determining our zeros so let's try two which is a factor of 12 so this would give us two to the third minus three times two squared minus four times two plus 12 which should be eight minus three times 4 is 12 minus 8 plus 12.

01:09 Lucky for us, this is equal to 0, which tells us x minus 2 is it term in the factored form of x to the third, or f of x, let's call it f of x to be short.

01:20 Now we're going to use polynomial long division, so we're going to have to say all of x over again.

01:26 So it's going to be x to the third minus 3x squared minus 4 x plus 12, and this will tell us the remainder the factored form of x.

01:39 I'm going to move this down a little bit.

01:42 Okay, and let's begin.

01:45 We know x goes into x to the third, x to the second time, x to the second times x is x to the third, x to the second times negative two is negative two, x squared.

01:57 When we subtract, we get negative x squared.

02:01 X goes into x squared, positive x times, positive x times x, oh wait, it goes in negative x times, because we want to negative.

02:11 Okay, negative x times x is equal to negative x squared, copying down this negative 4x, negative x times negative 2 is going to be positive 2x.

02:20 Once we subtract, that will give us negative 6x.

02:24 X goes into negative 6x, negative 6 times, negative 6 times x is negative 6x, copying down the 12...

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