In Exercises 83–94, find each product.

[5y + (2x+3)][5y − (2x+3)...

Question

In Exercises 83–94, find each product.

[5y + (2x+3)][5y − (2x+3)]

Accepted Answer

Hello, today we're going to be performing the multiplication on the given expression. So what we are given is the quantity nine Y plus four X plus 11 times nine Y minus the quantity four X plus 11. So we can go ahead and foil these two quantities together or we can go ahead and use the difference of squares to help us simplify this. Let's go ahead and take the approach of using the difference of squares. Now the difference of square states that if you have a binomial in the form of A squared minus B squared, then we can go ahead and rewrite this or factor this to be A plus B Times A. -7. And even though it's a bit hard to see at first, the given expression matches the right hand side of the difference of squares. See here, our A term is going to be nine Y and our B term is going to be four X plus 11. So we can go ahead and use the difference of squares to help us simplify this. But we're going to be using this property going from right to left. So we're going to allow our A to equal to nine Y. And we're going to allow B to equal four X plus 11. And if we put this in the form of the binomial A squared minus B squared, we can rewrite the original expression to be nine Y squared minus four X plus squared. Now I'm gonna go ahead and split this up into two parts. The left hand side is going to be part one and the right hand side is going to be part two. For part one, we have nine Y squared. And in order to simplify this, we're going to distribute the exponent of two to Y and nine, that is going to give us nine squared times Y squared and nine times nine is going to give us 81. So we have 81 Y squared. And for the second part, the second part gives us the quantity four X plus squared. Now, in order to simplify this, we can rewrite this as four X plus 11 times four X plus 11. And then we can go ahead and foil the two quantities together. So four X times four X is going to give us 16 X squared, 4 x times 11 is going to give us 44 x 11 times 4, X is going to give us 44 x again and 11 times 11 is going to give us 121. Now, we can go ahead and combine the 2 44 X terms together to give us 16 X squared plus 88 X plus 121. And that's what the second part is going to simplify too. So now that we have the simplified form of the first part and the second part, let's go ahead and replace this with the terms in the expression. So we can rewrite the expression as 81 Y squared minus the quantity 16 X squared plus 88 X plus 121. And finally, we can distribute negative one to the trinomial. And that is going to give us 81 Y squared minus X squared minus 88 X minus 121. And since we can't combine any of these terms together, this is going to be the simplified form of the given expression. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 83–94, find each product. [5y + (2x+3)][5y − (2x+3)]

Key Concepts

Factoring Difference of Squares

Distributive Property

Combining Like Terms

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