Question

Find the coefficient of the fourth term of (x+2)^5

•32
•48
•64
•80
Find the coefficient of the third term of (3x-1)^5
•248
•270
•360
•505
Find the coefficient of the third term of (a+5b^2)^4
•150
•500
•625
•1500

Answer 1

The coefficients of the binomial expansion
`(a+b)^n`, where n is the row number, is given in the Pascal's triangle shown below.

First, to find the coefficient of the fourth term of (x+2)^5 we look at row 5, term 4. The coefficient there is 10.

But, we must also remember that the term 2 also is taken to a certain power here. Mainly , for each term, the power of 2 is as follows:

2^0, 2^1, 2^2, 2^3=8.

So, in total we have: 10*8=80.


Second, to find the coefficient of the third term of (3x-1)^5 we again go to the row 5, this time term 3 and we have 10 there. Now we must check how each of (3x) and 1 expand, now being careful about the sign as well.

we have:
(3x)^5 (1) -(3x)^4 (1) (3x)^3(1)=27x^3.


Thus, the coefficient of the third term is 27*10=270.


Third, we want to find the coefficient of the third term of (a+5b^2)^4. We look at row 4, term 3. There we have 6.

The terms a and 5b^2 are as follows:

a^4 (5b^2)^0 a^3 (5b^2)^1 a^2 (5b^2)^2=25a^2b^4

Thus, the coefficient is 25*6=150.


Answer:

80; 270; 150