Question Video: Expanding a Binomial Using the Binomial Theorem | Nagwa Question Video: Expanding a Binomial Using the Binomial Theorem | Nagwa

# Question Video: Expanding a Binomial Using the Binomial Theorem Mathematics • Third Year of Secondary School

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Expand ((π₯/4) β (1/π₯))β΅.

04:10

### Video Transcript

Expand π₯ over four minus one over π₯ to the fifth power.

This is a binomial. Itβs the sum or difference of two algebraic terms. And weβre looking to raise it to the fifth power. Because weβre raising it to a nonnegative integer power, that means we can use the binomial theorem. This says that π plus π to the πth power, where π is a nonnegative integer, is π to the πth power plus π choose one π to the power of π minus one π and so on. Comparing our binomial to the general form, and we see weβre going to let π be equal to π₯ over four, π be equal to negative one over π₯, and π is the power, so itβs five.

The first term in our expansion is π to the πth power. So here, thatβs π₯ over four to the fifth power. Our next term is then five choose one times π₯ over four to the fourth power times negative one over π₯. Remember, we take π, and we reduce the power by one each time. But the power of π increases by one each time. So our next term is five choose two times π₯ over four cubed times negative one over π₯ squared. Weβre halfway there. We know that there will be π plus one, so six terms in our expansion. Letβs write out the remaining three.

They are five choose three times π₯ over four squared times negative one over π₯ cubed plus five choose four times π₯ over four times negative one over π₯ to the fourth power plus negative one over π₯ to the fifth power. Our job now is to simplify each of these terms. For our first term, thatβs fairly straightforward. We know that we can simply apply the fifth power to both parts of our fraction. So we get π₯ to the fifth power over four to the fifth power. And thatβs π₯ to the fifth power over 1024. Five choose one is simply equal to five. We know that this term is going to be negative since weβre multiplying by negative one over π₯. And then π₯ over four to the fourth power is π₯ to the fourth power over 256.

We then see that we can divide through by one power of π₯. So we get five times π₯ cubed over 256 times one, meaning our second term is negative five π₯ cubed over 256. The coefficient of our third term is five choose two. Now thatβs equal to 10. This time, weβre going to have a positive term since negative one over π₯ squared is simply positive one over π₯ squared. Similarly, π₯ over four cubed is π₯ cubed over 64. And now we might notice that we can divide through by π₯ squared and by two. 64 divided by two is 32. So we get five times π₯ over 32 times one, giving us five π₯ over 32.

Letβs keep going. Five choose three is once again 10. The coefficient of this term is going to be negative since negative one over π₯ cubed is negative one over π₯ cubed. And we get negative 10 times π₯ squared over 16 times one over π₯ cubed. And once again, we can divide through by π₯ squared. We can also divide through by two, giving us five and eight. So we get five times an eighth times one over π₯, which is five over eight π₯.

Thereβs two more terms to go. Five choose four is five. This time, weβre raising this negative term to an even power. So weβre going to get a positive result. And itβs five times π₯ over four times one over π₯ to the fourth power. And this time, we can also divide through by π₯. So we get five times a quarter times one over π₯ cubed. So this term is five over four π₯ cubed. The coefficient of our last term will be negative since weβre raising our negative value to an odd power. Itβs negative one over π₯ to the fifth power.

And so weβre finished with our binomial expansion. π₯ over four minus one over π₯ to the fifth power is equal to π₯ to the fifth power over 1024 minus five π₯ cubed over 256 plus five π₯ over 32 minus five over eight π₯ plus five over four π₯ cubed minus one over π₯ to the fifth power.

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