WEBVTT
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Determine the coefficient of π₯ to the power of negative six in the expansion of π₯ plus one over π₯ squared to the power of six.
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In this question, we have a binomial expansion written in the form π plus π to the πth power.
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We know that the general term denoted π sub π plus one is equal to π choose π multiplied by π to the power of π minus π multiplied by π to the power of π.
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We will begin by rewriting our expression as π₯ plus π₯ to the power of negative two raised to the sixth power.
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We can do this as one over π₯ to the power of π is equal to π₯ to the power of negative π.
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The general term of this expansion will therefore be equal to six choose π multiplied by π₯ to the power of six minus π multiplied by π₯ to the power of negative two to the power of π.
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This can be rewritten as six choose π multiplied by π₯ to the power of six minus π multiplied by π₯ to the power of negative two π.
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We know from our laws of exponents that when the base is the same, we can add the exponents or powers.
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Our expression becomes six choose π multiplied by π₯ to the power of six minus π plus negative two π.
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Simplifying this again gives us six choose π multiplied by π₯ to the power of six minus three π.
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In this question, weβre interested in the coefficient when the exponent of π₯ is negative six.
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We need to calculate the value of π when six minus three π equals negative six.
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We can subtract six from both sides of this equation so that negative three π is equal to negative 12.
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Dividing both sides by negative three gives us π is equal to four.
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We can now substitute this value back into the expression for our term.
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This gives us six choose four multiplied by π₯ to the power of negative six.
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The coefficient of this term is therefore equal to six choose four.
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We know that π choose π or πCπ is equal to π factorial divided by π minus π factorial multiplied by π factorial.
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Six choose four is therefore equal to six factorial divided by four factorial multiplied by two factorial.
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Six factorial is the same as six multiplied by five multiplied by four factorial.
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Canceling the four factorial gives us six multiplied by five divided by two factorial.
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As two factorial is equal to two and 30 divided by two is 15, six choose four is equal to 15.
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The coefficient of π₯ to the power of negative six in the expansion is 15.