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Expand negative eight π₯ minus two π¦ squared minus negative eight π₯ plus two π¦ squared.
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In this question, we have two binomials, negative eight π₯ minus two π¦ and negative eight π₯ plus two π¦.
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Weβre squaring each of them and then finding the difference.
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Letβs deal with squaring each binomial separately.
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And we can use two different methods to do this.
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What we must remember though is that when weβre squaring a binomial, weβre multiplying that binomial by itself.
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So in the case of negative eight π₯ minus two π¦ all squared, weβre looking for the result of negative eight π₯ minus two π¦ multiplied by negative eight π₯ minus two π¦.
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Weβll perform this expansion using the FOIL method.
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Remember, this is an acronym where each letter stands for a different pair of terms that we need to multiply together.
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F stands for firsts, so we multiply the first term in the first binomial by the first term in the second binomial.
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Thatβs negative eight π₯ multiplied by negative eight π₯.
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Negative eight multiplied by negative eight is 64.
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And π₯ multiplied by π₯ is π₯ squared.
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So we have 64π₯ squared.
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Next, O stands for outers or outside, so we multiply the terms on the outside of our binomials together.
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Thatβs the negative eight π₯ in the first binomial and the negative two π¦ in the second.
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Negative eight multiplied by negative two is positive 16 and π₯ multiplied by π¦ is π₯π¦.
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So we have positive 16π₯π¦.
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Next, we have I, which stands for inners or inside.
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So we multiply together the terms on the inside of the expansion.
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Thatβs negative two π¦ by negative eight π₯.
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Again, this gives positive 16π₯π¦.
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Finally, L stands for last, so we multiply the last term in each binomial together.
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Thatβs the negative two π¦ in the first by the negative two π¦ in the second, giving positive four π¦ squared.
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So weβve completed our expansion.
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And we notice that, at this point, we have four terms and the middle two terms are identical.
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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.
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For the second, negative eight π₯ plus two π¦ all squared, weβre looking for the result of multiplying negative eight π₯ plus two π¦ by itself.
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And this time, weβll use the distributive method.
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Weβll 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¦.
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And now, we just have to expand or distribute a single set of brackets or parentheses.
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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¦.
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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.
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As in our other expansion, we have four terms at this point, with two identical terms in the center.
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So we simplify to give 64π₯ squared minus 32π₯π¦ plus four π¦ squared.
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So weβve squared each binomial, and now weβre ready to perform the subtraction.
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We must be really careful here because we must make sure weβre subtracting every term in our second expansion from the first.
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We have 64π₯ squared plus 32π₯π¦ plus four π¦ squared minus 64π₯ squared minus 32π₯π¦ plus four π¦ squared.
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Now, of course, these two algebraic expressions are very similar because the binomials we started off with were very similar.
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They just differed in the sign of the π¦ term.
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So some of the terms will cancel when we subtract.
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But we must be very careful.
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We have 64π₯ squared minus 64π₯ squared, so those terms will cancel one another out.
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We also have four π¦ squared minus four π¦ squared.
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So those terms will also cancel each other out.
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What weβre left with is positive 32π₯π¦ minus negative 32π₯π¦.
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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¦.
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So we have our answer to the problem.
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Negative eight π₯ minus two π¦ all squared minus negative eight π₯ plus two π¦ all squared is equal to 64π₯π¦.
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And weβve seen two different methods for squaring binomials, the FOIL method and the distributive method.