Question

Find each coefficient described

7) 2nd term in expansion of (y − 2x)^4

8) 4th term in expansion of (4y + x)^4

9) 1st term in expansion of (a + b)^5

10) 2nd term in expansion of (y − x)^4

Answer 1

Formula : For the binomial expression (a + b)^n, The (r + 1) th term is

T_(r+1)  =  nCr X a^(n-r) X y^r -----------> (1)

 

7)

The expression is (y − 2x)^4

Comparing with binomial expression (a + b)^n

a = y,   b = -2x    and   n = 4

Toget 2nd term consider r = 1

Substitute corresponding values in Eq (!)

T_(1+1)  =  4C1 X (y)^(4-1) X (-2x)^1

T_2  =  [(4!)/(4-1)!(1!)] X y^3 X (-2x)

T_2  =  [(4!)/3!] X y^3 X (-2x)

T_2  =  - 2[(4 X 3 X 2 X 1)/(3 X 2 X 1)] X y^3 X (x)

T_2  =  - 2[(4)] X y^3 X (x)

T_2  =  - 8xy^3

 

8)

The expression is (4y + x)^4

Comparing with binomial expression (a + b)^n

a = 4y,   b = x    and   n = 4

Toget 4th term consider r = 3

Substitute corresponding values in Eq (!)

T_(3+1)  =  4C3 X (4y)^(4-3) X (x)^3

T_4  =  [(4!)/(1!)(3!)] X 4yx^3

T_4  =  [(4 X 3 X 2 X 1)/(3 X 2 X 1)] X 4yx^3

T_4  =  4 X 4yx^3

T_4  =  16yx^3

 

 

9)

The expression is (a + b)^5

Comparing with binomial expression (a + b)^n

a = a,   b = b    and   n = 5

Toget 4th term consider r = 0

Substitute corresponding values in Eq (!)

T_(0+1)  =  4C0 X (a)^(4-0) X (b)^0

T_1  =  1 X (a)^(4) X 1

T_1  =  a^4

 

10)

2nd term in expansion of (y − x)^4

Comparing with binomial expression (a + b)^n

a = y,   b = -x    and   n = 4

Toget 2nd term consider r = 1

Substitute corresponding values in Eq (!)

T_(1+1)  =  4C1 X (y)^(4-1) X (-x)^1

T_2  =  [(4!)/(4-1)!(1!)] X y^3 X (-x)

T_2  =  [(4!)/3!] X y^3 X (-x)

T_2  =  - [(4 X 3 X 2 X 1)/(3 X 2 X 1)] X y^3 X (x)

T_2  =  - [(4)] X y^3 X (x)

T_2  =  - 4xy^3

 

Answer :

7)  T_2  =  - 8xy^3

8)  T_4  =  16yx^3

9)  T_1  =  a^4

10)  T_2  =  - 4xy^3