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.

(3xy² − 4y)(3xy² + 4y)

Accepted Answer

Hello, today we're gonna be performing the multiplication on the given expression. So what we are given is six A B squared minus seven C times the quantity six A B squared plus seven C. Now there's two approaches that we can take, we can either foil the two quantities together or we can 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 squares states that if you have a binomial in the form of A squared minus B squared, then this can be factored to be A plus B Times A. -7. Now, in the given expression, the form of six A B squared minus seven C times six A B squared plus seven C matches the right hand side of the difference of squares. So we can go ahead and use the difference of squares by using the property going from right to left. So here we're going to allow A to equal to six A B squared and we're going to allow B to equal to seven C once we plug this into the form of the binomial for the difference of squares, we can rewrite this expression to be six A B squared raised to the power of two minus seven C raised to the power of two. Now, we're going to simplify both sides of this expression using our exponential distribution. We're going to distribute the exponent of two to B squared to A and to six. And we're going to do the same thing on the right hand side by distributing the exponent of two to C and seven doing this, we can go ahead and rewrite the expression to be six squared times A squared times B to the power of two squared, which is going to give us B to the power of four minus seven squared times C squared. Now, the only thing we have left to do is to simplify the coefficients six squared is the same thing as six times six, which is going to give us 36. And that's gonna be multiplied by A squared B to the fourth. And on the right hand side, seven squared is the same thing as seven times seven, which is going to give us 49. So we're going to be subtracting 49 C squared and 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 D. 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 69–82, multiply using the rule for the product of the sum and difference of two terms. (3xy² − 4y)(3xy² + 4y)

Key Concepts

Product of Sum and Difference

Binomial Expressions

Algebraic Simplification

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