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Use the binomial theorem to find the expansion of π minus π to the fifth power.
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π minus π is a binomial expression.
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Itβs an algebraic expression made up of two terms.
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Itβs raised to the fifth power.
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And weβre told weβre going to need to use the binomial theorem to find its expansion.
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And so, we recall that the binomial theorem says that for positive integers π, π₯ plus π¦ to the πth power is equal to the sum from π equals zero to π of π choose π times π₯ to the power of π minus π times π¦ to the πth power.
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Now, this expression can be quite nasty to work with, so we might consider its expanded form.
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This is π₯ to the πth power plus π choose one times π₯ to the power of π minus one times π¦ plus π choose two times π₯ to the power of π minus two times π¦ squared all the way through to π¦ to the πth power.
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Notice how the powers of π₯ reduce by one each time and the powers of π¦ increase.
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Letβs compare this formula to our binomial.
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We see that we can let π₯ be equal to π, π¦ is equal to negative π, and π, the exponent, is equal to five.
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This means the first term in our expansion is simply π to the fifth power.
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The second term is π choose one, so five choose one, times π to the power of five minus one or π to the fourth power times negative π.
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Remember, the powers of π₯, or the powers of π here, decrease by one each time, whereas the powers of π¦, which is negative π, increase by one each time.
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So, our third term is five choose two π cubed times negative π squared.
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Next, we have five choose three π squared times negative π cubed.
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Our fifth term is five choose four π times negative π to the fourth power.
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And our final term is negative π to the fifth power.
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Weβre going to evaluate five choose one, five choose two, five choose three, and five choose four.
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And so, we recall the formula to help us evaluate π choose π.
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Itβs π factorial over π factorial times π minus π factorial.
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This means five choose one is five factorial over one factorial times five minus one factorial or five factorial over one factorial times four factorial.
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Next, we recall that five factorial is five times four times three times two times one.
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Similarly, four factorial is four times three times two times one.
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And we see that we can divide through by four, three, two, and one.
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In fact, what weβre really doing is dividing through by four factorial.
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And so, we find five choose one is simply five divided by one, which is five.
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In much the same way, five choose four also yields a result of five.
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Letβs evaluate five choose two.
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Itβs five factorial over two factorial times five minus two factorial.
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Thatβs five factorial over two factorial times three factorial.
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Letβs write five factorial this time instead of as five times four times three times two times one as five times four times three factorial.
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Then, we see we can divide through by three factorial.
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By writing two as two times one, we can see we can divide through by two.
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And five choose two is, therefore, five times two divided by one, which is 10.
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Five choose three is also 10.
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And so, weβre ready to find the expansion.
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Our first term is still π to the fifth power.
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Our next term is five π to the fourth power times negative π.
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So, thatβs negative five π to the fourth power π.
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Now, negative π squared is positive π squared.
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So, our third term is 10π cubed π squared.
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When we cube a negative number, we get a negative result.
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So, our fourth term is negative 10π squared π cubed.
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We then have five ππ to the fourth power.
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And when we raise a negative number to the fifth power, we get a negative result.
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So, our final term is π to the fifth power.
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And so, weβre done.
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Weβve used the binomial theorem to find the expansion of π minus π to the fifth power.
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Itβs π to the fifth power minus five π to the fourth power π plus 10π cubed π squared minus 10π squared π cubed plus five ππ to the fourth power minus π to the fifth power.