Question

task 1
applying the binomial theorem
in this activity, youll use the binomial theorem to find the expanded form of a binomial expression. answer the following questions using the binomial theorem.
$(a + b)^n = a^n + \frac{n}{1!}a^{n - 1}b^1 + \frac{n(n - 1)}{2!}a^{n - 2}b^2 + \frac{n(n - 1)(n - 2)}{3!}a^{n - 3}b^3 + \cdots + b^n$
question 1
use the binomial theorem to determine the expanded form of the binomial expression $(x + y)^4$.
$x^4 + y^4$ $x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$
$x^4 + 2x^3y + 3x^2y^2 + 2xy^3 + y^4$ $x^4 + x^3y + x^2y^2 + xy^3 + y^4$

Explanation:

Step1: Identify n, a, b

Here, $n=4$, $a=x$, $b=y$

Step2: Apply binomial theorem terms

First term: $a^n = x^4$
Second term: $\frac{n}{1!}a^{n-1}b^1 = \frac{4}{1}x^{3}y = 4x^3y$
Third term: $\frac{n(n-1)}{2!}a^{n-2}b^2 = \frac{4\times3}{2}x^{2}y^2 = 6x^2y^2$
Fourth term: $\frac{n(n-1)(n-2)}{3!}a^{n-3}b^3 = \frac{4\times3\times2}{6}x^{1}y^3 = 4xy^3$
Fifth term: $b^n = y^4$

Step3: Sum all terms

Add the calculated terms together.

Answer:

$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$