Find each product. See Example 5.(x + 1) (x + 1) (x - 1) (x - 1)

Question

Accepted Answer

Hey, everyone here we are asked to perform the indicated operations to simplify the given expression where we have the quantity of Q plus three multiplied by the quantity of Q plus three multiplied by the quantity of Q minus three multiplied by the quantity of Q minus three. Here we have four answer choice options. Answer choice A Q raise to the fourth power plus six QQ minus 54 Q minus answer BQ raise to the fourth power minus six QQ plus 54 Q minus 81. Answer CQ raise to the fourth power minus 18 Q squared plus and answer DQ raised to the fourth power plus 18 Q squared minus 81. So to begin simplifying our given polynomial expression, we first want to rearrange it into two of the same group. So our first group will contain the quantity of Q plus three multiplied by the quantity of Q minus three. And now this first group is multiplied by our second group which contains the same sets of parentheses. So it also has the quantity of Q plus three multiplied by the quantity of Q minus three. And now with our two groups, we see that they individually follow the difference of squares formula. So to begin our first simplification, we need to recall the difference of squares formula, which states that the quantity of X plus Y multiplied by the quantity of X minus Y is equivalent to X squared minus Y squared. And now we just want to compare the left side of our formula to our individual groups so that we can identify our values for X and Y. So starting with our value for X, we can see that the term in the first term in each set of parentheses in our given expression is Q. So X is going to be Q and that means Y is going to be three. And so now we just want to find our X squared and Y squared shown on the right side of our formula. So X squared is just going to give us Q squared and then Y squared is just three squared and three squared simplifies to nine. So now we just want to substitute in this given information. So we can simplify both of our groups. So our first group, again, we recall that it is the quantity of X plus three multiplied by the quantity of X minus three. Well, from our formula, we know this expression is equivalent to Q squared minus nine. And now we can repeat this with our second group which follows the same pattern. So it also simplifies to Q squared minus nine. So now we have the quantity of Q squared minus nine multiplied by the quantity of Q squared minus nine. And now, since we have two of the same expression, we can simplify by rewriting it as the quantity of Q squared minus nine and this quantity is squared. So now with our simplified expression, we can continue to further simplify it by recalling our second formula where we have the quantity of X minus Y squared is equivalent to X squared minus two XY plus Y squared. And so now we want to find our new X and Y values by comparing our second formula with our current equation or expression. So doing this, we see our first term is Q squared. So X is going to be Q squared and that means Y is going to be nine. And now we just want to start finding the terms on the right side of our formula. So we can start with X squared which gives us the quantity of Q squared squared. And this simplifies two Q raised to the fourth power. And then we have Y squared which gives us nine squared which simplifies 2 81. And so now we just want to substitute in our information that we have into our second formula. So we see that the quantity of Q squared minus nine squared is equivalent to X squared, which we found to be Q raised to the fourth power we then have minus two multiplied by Q squared multiplied by nine. And lastly, we have plus 81. So now we just need to finish simplifying our expression by performing the necessary multiplication steps. So here our expression becomes Q raised to the fourth power minus 18 Q squared plus 81. And with our final simplified expression, here we are left with answer choice C where again our simplified expression is Q raised to the fourth power minus 18 Q squared plus 81. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Find each product. See Example 5.(x + 1) (x + 1) (x - 1) (x - 1)

Key Concepts

Polynomial Multiplication

Difference of Squares

Combining Like Terms

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