Question

Which of the following statements regarding the expansion of (x + y)^n are correct?
A. The coefficients of xn and y^n both equal 1.
B. For any term x^ay^b in the expansion, a + b = n.
C. For any term x^ay^b in the expansion, a - b = n.
D. The coefficients of x^ay^b and x^by^a are equal.

Answer 1

Answer:

A, B and D are true statements.

Step-by-step explanation:

We are given a binomial expansion

$(x+y)^n=^nC_0x^ny^0+^nC_1x^{n-1}y^1+^nC_2x^{n-2}y^2+...........+^nC_nx^0y^n$

$(x+y)^n=x^n+nx^{n-1}y+^nC_2x^{n-2}y^2+...........nxy^{n-1}+y^n$

Now we will check each option

Option A: The coefficients of $x^n$ and $y^n$ both equal 1.

If we see first and last term of the expansion, This statement is true.

Option B: For any term $x^ay^b$ in the expansion, a + b = n.

Let we take 3rd term of expansion $^nC_2x^{n-2}y^2$

Here, a=n-2 and b=2

If we do  a+b = n-2+2=n

a+b=n is true statement.

Option C: For any term x^ay^b in the expansion, a - b = n.

Let we take 3rd term of expansion $^nC_2x^{n-2}y^2$

Here, a=n-2 and b=2

If we do  a-b = n-2-2=n-4≠n

a-b=n is false statement.

Option D: The coefficients of x^ay^b and x^by^a are equal.

If we take second term from beginning and last of the expansion.

$\text{Coefficient From beginning } nx^{n-1}y=n$

$\text{From last } nxy^{n-1}=n$

This statement true.

Answer 2

By using what we know about expansions, we will see that A, B, and D are true.

How to expand exponential expressions?

Here we have the expression (x + y)^n

This means that we multiply the polynomial (x + y) by itself n times.

Now let's analyze each statement.

A) This is true, there is only one x in the original polynomial, and we multiply it by itself n times, then we can have only one term equal to x^n (or y^n)

B) True, if we multiplicate the variables by themselves n times, then the degree of the term must also be n.

Remember that for a term:

(x^a)*(y^b) the degree is a + b.

Then we always must have that a + b = n.

C) Because of what we wrote above, this is false.

D) This is true, because in the original polynomial the coefficients of y and x are the same, then in the expansion the coefficients of the reflected terms like (x^a)*(y^b) and (x^b)*(y^a) are equal.

If you want to learn more about polynomials you can read:

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