Question Video: Expanding Algebraic Expressions Using Algebraic Identities

Expand (βˆ’8π‘₯ βˆ’ 2𝑦)Β² βˆ’ (βˆ’8π‘₯ + 2𝑦)Β².

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Video Transcript

Expand negative eight π‘₯ minus two 𝑦 squared minus negative eight π‘₯ plus two 𝑦 squared.

In this question, we have two binomials, negative eight π‘₯ minus two 𝑦 and negative eight π‘₯ plus two 𝑦. We’re squaring each of them and then finding the difference. Let’s deal with squaring each binomial separately. And we can use two different methods to do this. What we must remember though is that when we’re squaring a binomial, we’re multiplying that binomial by itself. So in the case of negative eight π‘₯ minus two 𝑦 all squared, we’re looking for the result of negative eight π‘₯ minus two 𝑦 multiplied by negative eight π‘₯ minus two 𝑦.

We’ll perform this expansion using the FOIL method. Remember, this is an acronym where each letter stands for a different pair of terms that we need to multiply together. F stands for firsts, so we multiply the first term in the first binomial by the first term in the second binomial. That’s negative eight π‘₯ multiplied by negative eight π‘₯. Negative eight multiplied by negative eight is 64. And π‘₯ multiplied by π‘₯ is π‘₯ squared. So we have 64π‘₯ squared. Next, O stands for outers or outside, so we multiply the terms on the outside of our binomials together. That’s the negative eight π‘₯ in the first binomial and the negative two 𝑦 in the second. Negative eight multiplied by negative two is positive 16 and π‘₯ multiplied by 𝑦 is π‘₯𝑦. So we have positive 16π‘₯𝑦.

Next, we have I, which stands for inners or inside. So we multiply together the terms on the inside of the expansion. That’s negative two 𝑦 by negative eight π‘₯. Again, this gives positive 16π‘₯𝑦. Finally, L stands for last, so we multiply the last term in each binomial together. That’s the negative two 𝑦 in the first by the negative two 𝑦 in the second, giving positive four 𝑦 squared. So we’ve completed our expansion. And we notice that, at this point, we have four terms and the middle two terms are identical. We simplify by collecting the like terms, giving 64π‘₯ squared plus 32π‘₯𝑦 plus four 𝑦 squared for the result of expanding the first binomial.

For the second, negative eight π‘₯ plus two 𝑦 all squared, we’re looking for the result of multiplying negative eight π‘₯ plus two 𝑦 by itself. And this time, we’ll use the distributive method. We’ll take one of our factors of negative eight π‘₯ plus two 𝑦 and distribute it over the other, giving negative eight π‘₯ multiplied by negative eight π‘₯ plus two 𝑦 plus two 𝑦 multiplied by negative eight π‘₯ plus two 𝑦. And now, we just have to expand or distribute a single set of brackets or parentheses.

For the first set, we have negative eight π‘₯ multiplied by negative eight π‘₯, giving 64π‘₯ squared, and then negative eight π‘₯ multiplied by positive two 𝑦, giving negative 16π‘₯𝑦. And then, we have positive two 𝑦 multiplied by negative eight π‘₯, giving negative 16π‘₯𝑦, and positive two 𝑦 multiplied by positive two 𝑦, giving positive four 𝑦 squared. As in our other expansion, we have four terms at this point, with two identical terms in the center. So we simplify to give 64π‘₯ squared minus 32π‘₯𝑦 plus four 𝑦 squared.

So we’ve squared each binomial, and now we’re ready to perform the subtraction. We must be really careful here because we must make sure we’re subtracting every term in our second expansion from the first. We have 64π‘₯ squared plus 32π‘₯𝑦 plus four 𝑦 squared minus 64π‘₯ squared minus 32π‘₯𝑦 plus four 𝑦 squared. Now, of course, these two algebraic expressions are very similar because the binomials we started off with were very similar. They just differed in the sign of the 𝑦 term. So some of the terms will cancel when we subtract. But we must be very careful.

We have 64π‘₯ squared minus 64π‘₯ squared, so those terms will cancel one another out. We also have four 𝑦 squared minus four 𝑦 squared. So those terms will also cancel each other out. What we’re left with is positive 32π‘₯𝑦 minus negative 32π‘₯𝑦. Now, because we are subtracting a negative, those two signs together will form a positive, giving us 32π‘₯𝑦 plus 32π‘₯𝑦, which is 64π‘₯𝑦. So we have our answer to the problem. Negative eight π‘₯ minus two 𝑦 all squared minus negative eight π‘₯ plus two 𝑦 all squared is equal to 64π‘₯𝑦. And we’ve seen two different methods for squaring binomials, the FOIL method and the distributive method.

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