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'm so glad you're here. We are given this equation. We have six Y to the fourth equals 1296. Why? And we are to solve for why? Well what I want to do first is get both of these terms on the same side so that the right hand side will be set equal to zero and I'm bringing them over to the left because that will keep our six wide of the fourth or the term with the higher degree positive. And that makes it a little bit easier for factoring. So we've got six y to the fourth minus Y equals zero. And now we ask ourselves is there anything that can be factored out of both of those two terms? And yeah, they are both divisible by A six Y. So we can factor six Y at a bolt six Y to the fourth divided by six Y. Well that's Y to the third minus and 1296 Y divided by six is going to give us while negative. 216. So we've got Y to the third minus 216. And we noticed that that wide of the 3rd -216. Well those are both cubes. Why do the 3rd is why cubed and negative? 216? That is six cubed. So when we have a difference of cubes, we recall from previous lessons that we can use this formula that A to the third minus B. To the third equals parentheses A minus B times parentheses A squared plus A. B plus B squared. And we identify that are A. Is Y and R. B is six. So now we can plug in for R. A. S. And Arby's. So carrying down R six Y. And we've got two cents of parentheses now. And for the first one it's a minus B. So that's why minus six A squared. Is y squared plus A times B. Well that's six. Y plus B squared. That's six squared is 36. Great. Now we've got three factors and we can set each one of them equal to zero. We have six Y equals zero. We have y minus six equals zero and we have y squared plus six. Y plus 36 equals zero. For six, Y equals zero. To solve for Y. We divide both sides by six and zero divided by six. Well that is zero. So one of our wise is zero part of that zero product principle. For the Y -6 equals zero. We add 6 to both sides. And so we get y equals six as another one of our wide solutions. And now for this last one, y squared plus six, Y plus 36 equals zero. We can't factor that one very easily but we can use the quadratic formula. So we first want to identify what our A R B and our C terms. So A is in front of the first one. So A. Is one. B is in front of the second term that is six and C is in front. Well CSR constant. So C equals 36. And the quadratic formula is as we recall from previous lessons negative B plus or minus. The square root of B squared minus four times A times C. All over two A. So let's fill in our A's and B's and C's. So we have negative six plus or minus. The square root of six squared minus four times one times 36. All over two times 1. Simplifying further we have negative six plus or minus. The square root of six squared is 36 minus four times one times 36 is 144. All over two times one is two. Inside the radical we've got negative six plus or minus. And excuse me now. Inside the radical -144 is negative. 108. All over two. Well we know that the square root of -108, That's going to give us an eye out in front so that it's the square root of 108. And the square root of 108 is going to be a six. So six I square root of three. So now we've got negative six plus or minus six. I square root of three. All over two. And we can divide negative six and six I squared of three by two. So that will give us a negative three plus or minus three I square root of three. We can rewrite that as these two answers, we get y equals negative three plus three I squared of three and we get y equals negative three minus three I square root of three. And those are our four solutions. We had a degree of force. So that makes sense. We would be looking for four solutions. We look at our answer choices and this matches with answer choice D. Well done, we'll catch you on the next one.

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