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)^9; fifth term

Accepted Answer

Hey, everyone here, we are asked to perform binomial expansion on the expression, the quantity of D minus three to the seventh power to find the fourth term. Now, before we start expanding the entire expression, we need to recall the formula for binomial expansion which states that the quantity of A plus B to the NTH power is equal to the sum from R is equal to zero to N of NR times A to the power of N minus R times B to the power of R. So now looking at this formula, we can see that our first term begins when R is equal to zero, which means that our fourth term is when R is equal to three. And now we can use this right hand side of the formula to write out our fourth term. So we want to identify the rest of our variables here. So starting with the A term in our formula and looking at our given expression, we see that A is just D and then we see that B is this negative three. And lastly, our bi polynomial is raised to the seventh power, which means that our N has the value of seven. So with all of this information identified, we can just plug it into our formula and find our fourth term. So beginning with our coefficient, our topmost value is N. So we have seven and then our lower value is R. So we have three times A which again is D to the power of N minus R which is seven minus three times B which again is negative three to the power of R which is three. So now we just need to start simplifying. And with this coefficient here, we need to recall how to calculate this kind of coefficient for a binomial. So this form nr is equal to N factorial divided by the quantity of N minus R factorial times R factorial. So now we have already identified the N and R value. So we can just plug them into this formula to find the value of our coefficient. So beginning with the enumerator, we have seven factorial then divided by the quantity of seven minus three factorial times three factorial. And then our next term here we have D to the power of seven minus three which simplifies to D to the power of four. And then we have times negative three to the power of three. So now we can further simplify this looking at our fraction. Here we have seven factorial divided by seven minus three, gives us four factorial times three factorial. Then we have times our next term D to the fourth power times negative three to the power of three. And then finally simplifying all these terms, we see that our fourth term comes out to negative 945 times D to the fourth power, which is our answer here. For choice. D. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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

Key Concepts

Binomial Theorem

Binomial Coefficients

Term Position in Expansion

Watch next