Video Transcript
Consider the binomial expansion of two 𝑥 minus 𝑦 to the ninth power in ascending powers of 𝑥. What is the seventh term?
We recall that the binomial expansion of 𝑎 plus 𝑏 to the 𝑛th power is as shown. The general term is equal to 𝑛𝐶𝑟 multiplied by 𝑎 to the power of 𝑛 minus 𝑟 multiplied by 𝑏 to the power of 𝑟, where 𝑛𝐶𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial multiplied by 𝑟 factorial. In this expansion, the powers of 𝑎 are descending, whereas the powers of 𝑏 are ascending.
In this question, we’re asked to write the expansion in ascending powers of 𝑥. In order to do this, we can rewrite two 𝑥 minus 𝑦 to the ninth power as negative 𝑦 plus two 𝑥 to the ninth power. The terms two 𝑥 and negative 𝑦 swap places. The general term is denoted 𝑎 sub 𝑟 plus one. In this question, we’re asked to find the seventh term which is denoted 𝑎 sub seven. This means that the value of 𝑟 is six. The power or exponent 𝑛 is equal to nine. The first term in the parentheses, which will have descending powers, is negative 𝑦. And the second term, which will have ascending powers, is two 𝑥. The seventh term 𝑎 sub seven is therefore equal to nine 𝐶 six multiplied by negative 𝑦 to the power of nine minus six multiplied by two 𝑥 to the power of six.
Nine 𝐶 six or nine choose six is equal to nine factorial divided by nine minus six factorial multiplied by six factorial. We can rewrite nine factorial as nine multiplied by eight multiplied by seven multiplied by six factorial. As nine minus six is equal to three, the denominator is three factorial multiplied by six factorial. We can then divide the numerator and denominator by six factorial. Three factorial is equal to six. So we are left with nine multiplied by eight multiplied by seven all divided by six, which is equal to 84. Nine choose six is equal to 84.
Nine minus six is equal to three, and negative 𝑦 all cubed is the same as negative one cubed multiplied by 𝑦 cubed. This is negative 𝑦 cubed. Raising two 𝑥 to the sixth power is the same as two to the sixth power multiplied by 𝑥 to the sixth power. This is equal to 64𝑥 to the sixth power. The seventh term is therefore equal to 84 multiplied by negative 𝑦 cubed multiplied by 64𝑥 to the sixth power. 84 multiplied by 64 is 5376. This means that the seventh term of the binomial expansion two 𝑥 minus 𝑦 to the ninth power in ascending powers of 𝑥 is negative 5376𝑥 to the sixth power 𝑦 cubed.
