WEBVTT
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Find the coefficient of π₯ cubed in the expansion of two plus three π₯ all to the eighth power.
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This is an example of a binomial expression written in the form π plus π all raised to the πth power.
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We can solve problems of this type using Pascalβs triangle.
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However, as the power or exponent in this case is larger than five, it would be very time-consuming to write out every row of Pascalβs triangle.
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We will therefore use the binomial theorem.
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This states that π plus π to the πth power is equal to π choose zero multiplied by π to the πth power plus π choose one multiplied by π to the π minus oneth power multiplied by π to the first power, and so on, where π choose π is equal to π factorial divided by π minus π factorial multiplied by π factorial.
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As we move along term by term, the powers or exponents of π decrease, whereas the powers or exponents of π increase.
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In this question, the value of π is two, π is equal to three π₯, and the power or exponent π equals eight.
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At this stage, we could write out the whole expansion.
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However, we are only interested in the term containing π₯ cubed.
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As π is equal to three π₯, this will be the term in the general expansion containing π cubed.
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This term is equal to eight choose three multiplied by π to the fifth power multiplied by π cubed.
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Substituting in our values of π and π, we have eight choose three multiplied by two to the fifth power multiplied by three π₯ all cubed.
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Eight choose three is equal to eight factorial divided by five factorial multiplied by three factorial.
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We recall that eight factorial can be rewritten as eight multiplied by seven multiplied by six multiplied by five factorial.
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We can then cancel five factorial from the numerator and denominator.
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Three factorial is equal to six.
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We can therefore cancel this from the numerator and denominator.
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Eight choose three is therefore equal to eight multiplied by seven, which is equal to 56.
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Alternatively, we couldβve just typed this straight in to our calculator.
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Two to the fifth power is equal to 32.
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As three cubed is equal to 27, three π₯ all cubed is 27π₯ cubed.
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Our expression becomes 56 multiplied by 32 multiplied by 27π₯ cubed.
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This is equal to 48,384π₯ cubed.
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As we just want the coefficient of π₯ cubed, the final answer is 48,384.