How do i do this question using the binomial theorem concept? It says,
expand (2+x)^5 up to x^2 term, hence, expand (2 + x + x^2)^5 up to the x^2 term.
I could do the first part, and the answer would be (32 + 80x + ...)
But how should i make use of the answer i got in the second part?
please help!
Do the second part this way:
((2+x) + x^2)^5
you get the x^2 term really fast.
To expand (2+x)^5 up to the x^2 term, you can use the Binomial Theorem. The Binomial Theorem states that the expansion of (a + b)^n can be expressed as the sum of the terms of the form:
C(n, k) * a^(n - k) * b^k,
where C(n, k) is the binomial coefficient and is given by:
C(n, k) = n! / (k! * (n - k)!)
Here, n represents the power or the degree of the binomial and k represents the position of the term in the expansion.
Now, for the first part of the question, we have (2 + x)^5. Using the binomial theorem, we can expand it up to the x^2 term. The term that contains x^2 will be when k = 2 in the above formula.
So, let's apply the formula for (2 + x)^5 up to the x^2 term:
C(5, 2) * 2^(5 - 2) * x^2 = 10 * 2^3 * x^2 = 80x^2
Hence, the expansion of (2 + x)^5 up to the x^2 term is:
32 + 80x + 80x^2 + ...
Now, for the second part of the question, we need to expand (2 + x + x^2)^5 up to the x^2 term. To do this, we can simply substitute the answer from the first part into this expression.
So, the expansion of (2 + x + x^2)^5 up to the x^2 term will be:
(32 + 80x + 80x^2 + ...)^5
Keep in mind that we are only interested in the term up to x^2, so any terms after that can be represented as "..." or ignored.