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.

(y³ + 2)(y³ − 2)

Accepted Answer

Hey, everyone in this problem, we're asked to perform multiplication on the given expression. The expression we're given is two binomials multiplied together. So we have the first binomial W cubed plus five. In the 2nd Binomial WQ -5, we're gonna multiply them together. We're given four answer choices. Option A W to the exponent six minus 25. Option B W to the exponent six minus 10, W plus 25. Option C W to the exponent 6 plus 25. And option D W to the Exponent 6 -10, W -25. We're gonna start by rewriting the expression we were given. So we have the binomial W cubed plus five multiplied by the Binomial W. Cubed -5. And we can notice that this is actually a difference of squares. So we recall a difference of squares, we have the two binomials A plus B multiplied by A minus B. We can write this as the difference of squares A squared minus B squared. So the expression we were given WQ plus multiplied by WQ -5 is exactly the left hand side of the difference of squares equation where A is equal to W Q and B is equal to five. That means that we can write it as the right hand side of our difference of squares equations A squared minus B squared. And that's a simple way to get a resulting expression from the multiplication. So our original expression W cubed plus five a multiplied by W cubed minus five is going to be equal to A squared. In our case A is W Q. So we have W Q squared minus B squared five squared. Now the multiplication step is done and we're left with simplifying these exponents. So here we have W cubed to the exponent two. So we have an exponent race to an exponent recall that if we have an exponent, so say X to the exponent M raise to the exponent end, we can write this as X to the exponent M multiplied by N OK. So an exponent raised to an exponent, we can combine nodes into one single exponent by multiplying them together. So W cubed to the exponent two can be written as W to the exponent six -5 Squared. So -25 and that's the result of our multiplication. That is a simplified expression, we can't simplify any further. Now, if you didn't recognize right off the bat that this was a difference of squares, you can go ahead and multiply it. However, you'd like OK, we have two binomials multiplied together. So there are plenty of ways to multiply that, for example, you could use the foil method and work through the foil method and you're gonna get the exact same result. All right. So this, recognizing it's a different difference of squares, saves you a little bit of time. It saves you a little bit of that messy work in between where you can have some sign errors and things that don't work out so nicely. Um But you will get the exact same result. So we performed multiplication and we found that the result is W to the exponent 6 -25, which corresponds with answer choice. A 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. (y³ + 2)(y³ − 2)

Key Concepts

Product of Sum and Difference

Polynomial Multiplication

Simplifying Expressions

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