Solve each polynomial equation in Exercises 1–10 by factoring and...

Question

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.

3x^4 = 81x

Accepted Answer

Welcome back. I am so glad you're here. We are. Given this polynomial equation six Y to the fourth equals Y. And were asked to self for why? So I am going to start off by bringing this 1296 Y. To the left hand side, subtracting it from both sides of the equation so that this will be equal to zero. So on the left will have six Y to the fourth minus 1296. Y. On the right hand side equals to zero. Now we can do a little bit of factoring. Is there anything that those two terms have in common that we could factor out? Yes. They both have a six Y in them. So the first term, six Y to the fourth divided by six Y leaves us with Y to the third and then minus 1296. Y divided by six Y. Well that's a negative 216. That is equal to zero. All right. We've got one thing factored out. We look at what's left inside the parentheses and we notice that that is a difference of cubes. What are the two cubes? Well, the first one is why cubed. And the second one is six cubed. We recall from previous lessons that when we have a difference of cubes we can identify our a term in our B term and then what we'll have is we have this a cute minus be cute is equal to factoring it out A minus B parentheses. Times another parentheses A squared plus A. B plus B squared. So what are our A. And R B. Well R A. Is Y and R B is six. So now we can plug those in for this right hand side of the equation A minus B. That's why minus six. And then new parentheses A squared is y squared plus A times B. That's why times six or six Y plus B squared B squared will be is six. So that's six squared which is 36. And now we can plug those back in over here. So we've got six Y times now Y minus six times parentheses Y squared plus six. Y plus 36. All equal to zero. Now we've got three factors and we can start to solve for y for each of those factors. For the first one we have six. Y we can set that equal to zero, divide both sides by six and we get y equals zero, divided by six is zero. There's one of our solutions, Y equals zero. All right, the next one we can set why minus six equal to zero for that one. We can add six to both sides and then we'll have Y equals on the left zero plus six is six. There's another of our Y solutions. Great. We've got two of them. Now we have this Y squared plus six. Y plus 36 equals zero. And this one we can't factor it but we can use the quadratic formula. So the quadratic formula as we recall from previous lessons is negative B plus or minus the square root of B squared minus four A. C. All divided by two A. Now what are our? A. B and C terms are A. Is in front of the Y squared R. A. Is one. R. B. Is in front of the Y. That is six and R. C. Is the constant term. That is 36. So now let's plug in our A. B. And C terms will have negative not be but six plus or minus. The square root of B is six squared minus four times A. Is one times C. Is 36. All over two times A. Which is one. Alright let's simplify where we can mostly inside the radical there. So bring down that minus six plus or minus inside the radical will do six squared is 36 minus four times one times 36. That is minus 144. All over two times 12 times one is two. Okay simplifying inside the radical some more. We have bringing down our negative six plus or minus minus 144 is 108. Excuse me? It's negative 108. So we've got negative 108. Inside the radical. Alright we can do a bit of work on the square root of negative 108. Okay we've got what? How can we break down negative 108. Well, negative 108 is the same thing is negative one times 108. And we know from previous lessons that by definition the square root of negative one is equal to I. So we can bring out this square root of negative one and we can have an eye in front and that leaves us with I times the square root of 108. But we can break down 108 even further. We know that when we take 108 we can build a factor tree. We know it's divisible by two because it's even So that's two times 54 54 is nine times is three times three and six is two times three. So are there any factors that occur in twos that could come out? Well, we've got two twos. So inside of the radical, keeping that eye in front square root of we've got two squared and then we've also got two threes. So that's two squared times three squared. And then what's left over left over? Is this three? So that will be times three. And now anything that squared can come out two squared and three squared can come out. Well that will be two times three times I on the outside. Leaving on the inside of the radical. Just this three by itself, two times three is six. So we have six I times the square root of three. And that is the same thing as the square root of negative 108. All right now we've got negative six plus or minus instead of the square root of 100 negative 108. It's six I times the square root of three. All over two. Well, both of these here negative six and six I squared of three are divisible by two. So we'll divide both of those by two. That will be negative six divided by two is negative three plus or minus six. I squared of three, divided by two is three. I square root of 23. Not two squared of three. Okay, those are other two wise solutions after all that. We got two more. So we have Y equals the first one is negative three plus three I squared of three. And the next one is y equals negative three minus three I square root of three. And there are four solutions for y. And it makes sense. We started with a degree of four. So we've got four solutions. Now we look at our answer choices and some of them are very similar but we see that this C doesn't have a negative in front of that three there. And so this matches with answer choice. D Well done. We'll catch you on the next one

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. 3x^4 = 81x

Key Concepts

Polynomial Equations

Factoring

Zero-Product Principle

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