Solve each polynomial inequality in Exercises 1–42 and graph the ...

Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3−3x^2−9x+27<0

Accepted Answer

Hello today we're going to be solving the given polynomial inequality. So what we are given is x cubed minus four, X squared minus 16, X plus 60 for less than zero. Now, in order to solve for this inequality, I'm going to need a first factor the given polynomial completely. Since we are given four terms in the polynomial, we're going to factor this by using the method of factor by grouping. So I'm going to group up the first two terms and the last two terms together. And I'm going to go through each group and factor out a common factor. So in the first group, X cubed minus four, X squared has a common factor of X squared. So if I factor out X squared from the first group, what I am left with is x squared times the quantity x minus four. And in the second group 16 X and 64 are both divisible by 16. But I want this first term to be positive after the factory. So I'm gonna go ahead and factor out a negative 16 from the second group. And when I factor out a negative 16 from the second group, what I am left with is negative 16 times the quantity X minus four, less than zero. Now notice that we have two instances of x minus four. We can combine this into a single quantity X minus four by assigning the coefficients its own group And doing this is going to give us x squared minus 16 times the quantity X minus four, less than zero. Now there's one more thing that we can factor here, X squared minus 16 is factory ble because this is a difference of squares. And if we factor this using the difference of squares, what we are left with is X plus four times x minus four times the quantity X minus four because that was a previous factor and that's going to be less than zero. Now, next we have two instances of x minus four and we can go ahead and combine this into a single quantity by writing it as x minus four quantity squared. And that's going to be times X plus four because of the previous factor and this is going to be less than zero. Now this is the completely factored version of the original inequality. So now I want to find those regions of X on the number line that satisfy this inequality. But before I draw the number line, I'm going to need to figure out the zeros for X. In order to get the zeros for X, I'm going to take each factor of X. Set equal to zero and solve for X. So doing this is going to give me X plus four equal to zero and x minus four squared equal to zero on the left hand side. If I minus four to both sides of the equation, I get X is equal to negative four, which is going to be on my first zero on the number line. And on the right hand side I'm going to need to take the square root of both sides first. And that's going to give me x minus four is equal to the square root of zero, which is just zero. Now, if I add four to both sides of this equation, I get X is equal to positive four, which is the final zero that I need. So now let's go ahead and plot the zeros on the number line. So I'm gonna draw the number line and this number line is going to start a negative infinity. Going to our first year which is -4. Then from negative to positive four and finally ending from positive four to positive infinity. Now notice that the inequality is using the less than symbol, since it is using the less than symbol, we are not going to include the zeros in our inequality. So now that we have the regions of X, we need to take a testing point from each region of X, plug it into our inequality and whichever testing point satisfies the inequality is going to be the region that we include in our solution set. So let's go ahead and test the region to the left of -4. And this is going to be when X is equal to negative five. When x is equal to negative five, we get negative five plus four times the quantity negative five minus four squared. And this should be less than zero. Negative five plus four gives us negative one. And negative five minus four gives us negative nine squared. Now, negative nine squared is going to evaluate to give us positive positive 81 times negative one is going to give us negative 81 less than zero. And a negative number is always less than zero. So this inequality is true. So that means that since the testing point to the left of negative force satisfies the inequality we're going to be including the region that's to the left of negative four. Now let's pick a testing point between negative four and positive four. And that could be when X is equal to zero. Since zero lies between the two numbers. If we test X is equal to zero, we get zero plus four times the quantity zero minus four squared less than zero. Simplifying this is going to give us four times negative four squared to less than zero. Negative four squared is going to evaluate to give us positive 16 and 16 times four is going to give us 64. Less than zero. Now, a positive number can never be less than zero. So this is false. Since that is false, we are not going to include the region between negative four and four in our solution set. Finally, let's test the region to the right of positive four and that can be when X is equal to positive five. When x is equal to positive five we get five plus four times the quantity five minus four squared less than 05 plus four gives us 95 minus four is going to give us one squared less than 01 squared is just gonna be one and nine times one gives us nine. So we are left with nine less than zero which is false. So we're not going to include the region to the right of negative four. So now let's go ahead and interpret what we have on the number line. The number line is saying that we're going to include the region from negative infinity to negative four, not including four, negative four. And we're not including the region between -4 and after positive four And we are not including positive four in the solution set. So with that being said, when we want to write this answer, an interval notation. The region being included is going to start negative infinity. And since negative infinity doesn't have a finite value, we're going to assign it a parentheses. E Next this region ends at -4. And since -4 is not being included, we're going to assign it a parentheses as well. And no other region including positive four is going to be included in our solution set. So this is going to be the answer to the inequality. And with that being said, the answer to this problem is going to be be So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3−3x^2−9x+27<0

Key Concepts

Polynomial Inequalities

Graphing on a Number Line

Interval Notation

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