Question

Prove that the coefficient of $x^{n}$ in the expansion of $(1 +a)^{m+n}$ is twice the coefficient of $x^{n}$ in the expansion of $(1 + x)^{2n-1}$.

Answer 1

Example 1

Multiply the following monomials.

a) (2x

2

)(5x

3

)

b) (−3y

4

)(2y

2

)

c) (3xy5

)(−6x

4

y

2

)

d) (−12a

2b

3

c

4

)(−3a

2b

2

)

Solution

a) (2x

2

)(5x

3

) = (2 · 5)·(x

2

· x

3

) = 10x

2+3 = 10x

5

b) (−3y

4

)(2y

2

) = (−3 · 2)·(y

4

· y

2

) = −6y

4+2 = −6y

6

c) (3xy5

)(−6x

4

y

2

) = 18x

1+4

y

5+2 = −18x

5

y

7

d) (−12a

2b

3

c

4

)(−3a

2b

2

) = 36a

2+2b

3+2

c

4 = 36a

4b

5

c

4

To multiply a polynomial by a monomial, we use the Distributive Property.

This says tha