In Exercises 69–82, multiply using the rule for the product of th...

Question

In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.

(2 − y⁵)(2 + y⁵)

Accepted Answer

Hey, everyone in this problem, we're asked to perform multiplication on the given expression. The expression we given is two binomials multiplied together. The first is four minus P to the exponent six. And that's multiplied by four plus P to the exponent six. We're given four answer choices. Option A 16 plus eight P to the exponent six plus P to the exponent B 16 minus eight P to the exponent six plus P to the exponent 12. Option C 16 plus P to the exponent 12. In option D 16 minus P to the exponent 12. We're gonna start by rewriting the expression we were given. So we have the binomial four minus P to the exponent six multiplied by the binomial 4 Plus P to the exponent six. Before we start multiplying, we wanna notice that this is actually a difference of squares. And so recall a difference of squares. If we have a term A minus B and we multiply this to the binomial A plus B, you can write this as the difference of squares A squared minus B squared. If we compare this expression to our expression there, we have A minus B multiplied by A plus B, we can write that A, in our case is equal to four and B is equal to P to the exponent six. So we have the expression on the left hand side, we figured out what A and B are. And now we know that that's equivalent to this difference of squares on the right hand side. So that means we can write our expression four minus P to the exponent six multiplied by 4 Plus P to the exponent six is equal to A squared. In our case A is four. So we have four squared minus B squared, which is P to the exponent six squared. So now all we have left to do is to simplify these exponents for our expression four squared. And we know that that's 16, we have P to the exponent six squared. Well, let's recall what happens when we have an exponent squared if we have X to the exponent M and this is raised to the exponent end, we can write this as X to the exponent M multiplied by N OK. So the exponents multiply together. So in our case, we have P to the exponent six, all squared, we multiply those exponents and we get P to the exponent 12. And that's it. That's the result of the multiplication. Now, if you didn't recognize that this was a difference of squares, that's OK. You can go through the multiplication process whether you use the foil method uh or a different method. But whichever method you like to use for multiplying two binomials, you could work through that process and you would get the exact same result. If we compare this to our answer choices, we found that the result of the multiplication is 16 -2 to the exponent 12. This corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms. (2 − y⁵)(2 + y⁵)

Key Concepts

Product of Sum and Difference

Algebraic Expressions

Factoring and Expanding

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