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 to represent a set of numbers, indicating the beginning and end of the intervals where the inequality is true. It uses brackets to denote closed intervals (including the endpoint) and parentheses to denote open intervals (excluding the endpoint), effectively summarizing the solution set of the inequality.

Sign Analysis (Number Line Method)

Sign analysis involves using a number line to test the intervals created by the zeros of the polynomial. A sample point from each interval is substituted into the polynomial to determine whether the expression is positive or negative. This method systematically helps in identifying which intervals satisfy the given inequality.

Multiplicity and Graph Behavior

The multiplicity of a zero refers to the number of times a particular zero appears as a factor of the polynomial. It affects the graph’s behavior at that point; for example, zeros with an odd multiplicity cause the graph to cross the axis, whereas zeros with an even multiplicity cause the graph to touch the axis and turn around. This information is crucial in determining how the sign of the polynomial changes across the zeros.

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. They serve as critical points in solving polynomial inequalities since they divide the number line into intervals. By evaluating the sign of the polynomial in each interval, one can determine where the inequality holds true.

Polynomial Inequality

A polynomial inequality involves finding the set of real numbers that make a polynomial expression less than, greater than, or equal to zero. Solving these inequalities generally requires determining where the polynomial function is positive, negative, or zero, which can be done by analyzing its zeros and behavior on different intervals.

Transcript

00:01 Hello, today we will be solving a polynomial inequality.

00:03 This example asks us to find the values of x, which satisfies the inequality, x cubed plus x squared minus 5x plus 3 is less than or equal to 0.

00:14 So off the bat, let's start by making a function.

00:17 I'd like to call it f of x, and we'll make that equal to the left -hand side.

00:31 And we want this function to be less than or equal to zero.

00:36 So as usual, we're going to start by factoring this function.

00:40 So looking at this function, off the top of my head, there's no obvious way to factor it, and it's kind of hard because it's cubed.

00:47 So i'm going to start by plugging in random numbers to start to find a zero, and then from there i can divide to simplify it.

00:55 So you can see the three doesn't have a variable coefficient, so that means zero is not an option, because if x were zero, the y value of these three.

01:05 So let's start by plugging in 1.

01:10 So you're going to have 1 cubed plus 1 squared minus 5 times 1 plus 3.

01:18 That's just going to be 1 plus 1 minus 5 plus 3, negative 3 plus 3 plus 3 is 0.

01:25 So that was just a lucky first try.

01:28 So let's run with it.

01:29 So that tells us that x minus 1 is going to be one of the terms in the factored form of x cubed.

01:37 Plus x squared minus five x plus three okay so from here we'll do polynomial long division so we know x goes into x cubed x squared x squared times and from here we just multiply i'm going to draw a line to separate that out okay so we know x squared times x is going to be x cubed x squared times negative one is going to be negative x squared from here we'll subtract these two will cancel making zero and x squared minus negative x squared turns into a positive is going to be positive two x squared so how many times does x go into two x squared it goes into it two x times from here we're going to multiply again and we're going to drop down this negative 5x 2x 2x times x is 2x squared 2x times 2x squared minus 2x squared is 0 5x minus negative 2x is going to be negative 3x x x goes into negative 3x negative 3 times so we're going to we're going to drop this 3 down okay and negative 3 times x is going to be negative 3 and negative 3 times negative 1 is going to be positive 3 so that gives us the remainder 3 wait no it doesn't because i forgot to drop down the three earlier so negative oh positive three minus positive three is going to equal zero okay so this is even this isn't even dividend yes okay so that gives us the factored form f of x is equal to x minus one who's these x minus one time and then we'll just copy this master, times x squared plus 2x minus 3.

04:02 Okay, lucky for us, x squared plus 2x minus 3 is easily factorable, so that gives us a factored form, x minus 1 times x minus 1 times x plus 3.

04:18 And that's because 3 minus 1 is 2 and negative 1 times 3 is negative 3.

04:23 So to double check this, let's factor it.

04:26 Oh, well, we don't need to factor it out again because that would take us back to this form, which would be redundant.

04:33 Okay, so this factored form gives us our zeros...

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