Question

Hello,Could you help me with question 31? It is:Write the first three terms in each binomial expansion, expressing the result in simplified form

Answer 1

To solve this problem, we use the Binomial Theorem to compute the expansion:

`(a+b)^n=\sum_(k=0)^n\binom{n}{k}a^(n-k)b^k=\sum_(k=0)^n\frac{\text{ n!}}{(n-k)!k!}a^(n-k)b^k.`

Therefore, the first term of the expansion is:

`(8!)/((8-0)!0!)x^(8-0)2^0=(8!)/(8!1)x^8*1=x^8.`

Using the binomial theorem, we get that:

`(x+2)^8=x^8+16x^7+112x^6+448x^5+1120x^4+1792x^3+1792x^2+1024x+256.`

Therefore, the first three terms are:

`x^8,16x^7,112x^6.`

Answer:

`x^(8), 16x^(7), 112x^(6)`