Question Video: Expanding Algebraic Expressions Using Algebraic Identities | Nagwa Question Video: Expanding Algebraic Expressions Using Algebraic Identities | Nagwa

Question Video: Expanding Algebraic Expressions Using Algebraic Identities Mathematics • First Year of Preparatory School

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Expand (β8π₯ β 2π¦)Β² β (β8π₯ + 2π¦)Β².

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Video Transcript

Expand negative eight π₯ minus two π¦ squared minus negative eight π₯ plus two π¦ squared.

In this question, we have two binomials, negative eight π₯ minus two π¦ and negative eight π₯ plus two π¦. Weβre squaring each of them and then finding the difference. Letβ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 weβre squaring a binomial, weβre multiplying that binomial by itself. So in the case of negative eight π₯ minus two π¦ all squared, weβre looking for the result of negative eight π₯ minus two π¦ multiplied by negative eight π₯ minus two π¦.

Weβ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. Thatβ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. Thatβs the negative eight π₯ in the first binomial and the negative two π¦ in the second. Negative eight multiplied by negative two is positive 16 and π₯ multiplied by π¦ is π₯π¦. So we have positive 16π₯π¦.

Next, we have I, which stands for inners or inside. So we multiply together the terms on the inside of the expansion. Thatβs negative two π¦ by negative eight π₯. Again, this gives positive 16π₯π¦. Finally, L stands for last, so we multiply the last term in each binomial together. Thatβs the negative two π¦ in the first by the negative two π¦ in the second, giving positive four π¦ squared. So weβ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₯π¦ plus four π¦ squared for the result of expanding the first binomial.

For the second, negative eight π₯ plus two π¦ all squared, weβre looking for the result of multiplying negative eight π₯ plus two π¦ by itself. And this time, weβll use the distributive method. Weβll take one of our factors of negative eight π₯ plus two π¦ and distribute it over the other, giving negative eight π₯ multiplied by negative eight π₯ plus two π¦ plus two π¦ multiplied by negative eight π₯ plus two π¦. 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 π¦, giving negative 16π₯π¦. And then, we have positive two π¦ multiplied by negative eight π₯, giving negative 16π₯π¦, and positive two π¦ multiplied by positive two π¦, giving positive four π¦ 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₯π¦ plus four π¦ squared.

So weβve squared each binomial, and now weβre ready to perform the subtraction. We must be really careful here because we must make sure weβre subtracting every term in our second expansion from the first. We have 64π₯ squared plus 32π₯π¦ plus four π¦ squared minus 64π₯ squared minus 32π₯π¦ plus four π¦ 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 π¦ 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 π¦ squared minus four π¦ squared. So those terms will also cancel each other out. What weβre left with is positive 32π₯π¦ minus negative 32π₯π¦. Now, because we are subtracting a negative, those two signs together will form a positive, giving us 32π₯π¦ plus 32π₯π¦, which is 64π₯π¦. So we have our answer to the problem. Negative eight π₯ minus two π¦ all squared minus negative eight π₯ plus two π¦ all squared is equal to 64π₯π¦. And weβve seen two different methods for squaring binomials, the FOIL method and the distributive method.

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