Factor each polynomial by grouping. See Example 2.

Question

4 x^{6} + 36 - x^{6} y - 9 y

Accepted Answer

Hello. Everyone in this video. We're going to be looking at this practice problem where we want to factor the polynomial eight M. To the sixth plus 56 minus M. Two the sixth times N minus seven N. Using grouping. So in this case because we want to use grouping to factor, we're going to be grouping this polynomial into two groups, each containing two terms. So you can see that we have 1234 terms in this polynomial. So I'm going to make groupings of two and I want to try to make these groupings with terms that have similar components. So if I look at the terms, notice that the last two both have an end variable. So I'm going to try to keep those two terms together. Which means that my groupings will be the first two terms eight am to the sixth plus plus my second grouping negative. M. To the six times N minus seven N. And now from each of these groupings, I want to try to find the greatest common factor within those terms and factor it out. So if I look at this first grouping, you can see that the second term has no M. Variable. So I can't pull out an M. But I can't pull out an eight. So if I pull out an eight from eight AM to the sixth, I'm left with em to the sixth. If I pull out an eight from 56 I'm left with seven. So this first term is now eight times M two the sixth plus seven. And I can do the same for the second term where I want to pull out the greatest common factor. So you can see that they're both negative. So I can pull out a negative one. But I can also pull out this end variable because they both contain that end. So I can pull out a negative one end from this second term. If I do that I'm left with em to the sixth. This negative seven is now positive seven because I put the negative on the outside and now within these two terms or these two factors for the polynomial, you can see that it's the same M to the sixth plus seven. Which means that I can actually factor out em to the sixth plus seven once again. So when I factor that out, I'm left with eight minus. And so notice that these two equations are exactly the same. Because if I distribute the M. To the sixth plus seven to the eight and the negative N. I go back to the equation above. So that means that M. Two the six plus seven times eight minus N becomes our final factory ization for this polynomial. And you can see that that aligns with answer choice C. So hopefully this video helped and see you next time

Factor each polynomial by grouping. See Example 2. $4 x^{6} + 36 - x^{6} y - 9 y$

Key Concepts

Polynomial Grouping

Factoring Common Factors

Difference of Squares

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