In Exercises 39–48, find the term indicated in each expansion. (x...

Question

In Exercises 39–48, find the term indicated in each expansion. (x − 1/2)^9; fourth term

Accepted Answer

Hey everyone here we are asked to perform binomial expansion on the expression the quantity of a minus 8/9 to the power of five to find the second term. So before we start solving this problem here we need to recall the formula for binomial expansion, which states that the quantity of A plus B to the n power is equal to the sum from R is equal to zero to n. Of n R times A. To the power of n minus R times B to the power of our. And so here in this formula we can see that our begins at zero which means our first term is when R is equal to zero, telling us that our second term is when R is equal to one. So with this we can find the rest of our variables here. So starting with the A term in the formula while A. Is just A. In our expression so we have A. Is equal to A. Then for B. In our expression our B value is this negative 8/9 and then our end value is the power to which our binomial is raised to. So our end is equal to five. And now with this we can just plug in this information into our formula. So for the coefficient here we have this set of parentheses and the top value is five and the lower value is one times A. To the power of n minus R which is five minus one times B which is negative 8/ to the power of one. And now to simplify this expression here we need to recall how to calculate a binomial coefficient. So using this formatting this this coefficient of N. R is equal to N factorial divided by the quantity of n minus r. Factorial times R factorial. And with this we can plug in our values for N. And R. To find the value for our coefficient. So writing this out we have five factorial divided by the quantity of again and minus R. So we have five minus one factorial times one factorial and just rewriting the rest of our values here we have A to the power of five minus one and negative 8/9 to the power of one. So with this we can just continue to simplify. So again in our fraction here we have five factorial divided by and we have 5 -1 giving us four factorial times one factor. Then we have a to the power of five minus one which just gives us a to the power of four and then we have negative 8/9 to the power of one giving us just negative 8/9. And with this we can just simplify our coefficient here and we are left with our second term being negative A to the fourth power divided by nine which is our answer here for choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time

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In Exercises 39–48, find the term indicated in each expansion. (x − 1/2)^9; fourth term