Question Video: Finding a Certain Term in a Binomial Expansion Mathematics

Find the third term in the expansion of (2𝑥 + (5/√𝑥))⁵.

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Video Transcript

Find the third term in the expansion of two 𝑥 plus five over the square root of 𝑥 to the power of five.

This is an example of a binomial expansion written in the form 𝑎 plus 𝑏 to the power of 𝑛. We could write out the whole expansion. However, we know that the general term 𝑎 sub 𝑟 plus one is equal to 𝑛 choose 𝑟 multiplied by 𝑎 to the power of 𝑛 minus 𝑟 multiplied by 𝑏 to the power of 𝑟. In this question, we want to find the third term 𝑎 sub three. Inside the parentheses, we have two terms. 𝑎, the first term, is equal to two 𝑥. And 𝑏, the second term, is equal to five over root 𝑥.

The exponent or power that this is raised to is five. Therefore, 𝑛 is equal to five. As we are trying to find the third term, 𝑟 plus one is equal to three. Subtracting one from both sides of this equation gives us 𝑟 is equal to two. We can now substitute these four values into the general term formula. The third term is therefore equal to five choose two multiplied by two 𝑥 cubed multiplied by five over the square root of 𝑥 squared.

We know that 𝑛 choose 𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial multiplied by 𝑟 factorial. Five choose two is therefore equal to five factorial divided by three factorial multiplied by two factorial. We can rewrite five factorial as five multiplied by four multiplied by three factorial. This simplifies to five multiplied by four divided by two factorial, which equals 10. Five choose two is equal to 10.

When cubing two 𝑥, we can cube the two and 𝑥 separately. As two cubed is equal to eight, two 𝑥 cubed is equal to eight 𝑥 cubed. We can use a similar method when squaring a fraction. We square the numerator and denominator separately. Five squared is equal to 25, and the square root of 𝑥 squared is 𝑥. A square rooting and squaring are inverse operations. The third term is therefore equal to 10 multiplied by eight 𝑥 cubed multiplied by 25 over 𝑥.

This can be simplified to 2000𝑥 squared as 25 multiplied by eight multiplied by 10 is 2000 and 𝑥 cubed divided by 𝑥 is 𝑥 squared. The third term in the expansion of two 𝑥 plus five over the square root of 𝑥 to the power of five is 2000𝑥 squared. To save time, we could’ve also calculated five choose two on the calculator.

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