Let's break down the multiplication of the binomials [tex]\((x+y)^n\)[/tex] and identify the coefficients for each expansion.
1. For [tex]\((x+y)^1\)[/tex]:
Expanding [tex]\((x+y)\)[/tex], we get:
[tex]\[ x + y \][/tex]
So, the coefficients are:
[tex]\[ [1, 1] \][/tex]
2. For [tex]\((x+y)^2\)[/tex]:
Expanding [tex]\((x+y)^2\)[/tex], we get:
[tex]\[ (x+y)(x+y) = x^2 + xy + yx + y^2 \][/tex]
Combining like terms, we have:
[tex]\[ x^2 + 2xy + y^2 \][/tex]
So, the coefficients are:
[tex]\[ [1, 2, 1] \][/tex]
3. For [tex]\((x+y)^3\)[/tex]:
Expanding [tex]\((x+y)^3\)[/tex], we get:
[tex]\[ (x+y)(x+y)(x+y) \][/tex]
First, expand the first two binomials:
[tex]\[ (x+y)(x+y) = x^2 + 2xy + y^2 \][/tex]
Next, multiply this result by [tex]\((x+y)\)[/tex]:
[tex]\[ (x^2 + 2xy + y^2)(x+y) \][/tex]
[tex]\[ = x^3 + x^2y + 2x^2y + 2xy^2 + y^2x + y^3 \][/tex]
Combining like terms, we get:
[tex]\[ x^3 + 3x^2y + 3xy^2 + y^3 \][/tex]
So, the coefficients are:
[tex]\[ [1, 3, 3, 1] \][/tex]
Thus, the completed expansions and their coefficients are:
[tex]\[
(x+y)^1 = x + y
\][/tex]
Coefficients:
[tex]\[ [1, 1] \][/tex]
[tex]\[
(x+y)^2 = x^2 + 2xy + y^2
\][/tex]
Coefficients:
[tex]\[ [1, 2, 1] \][/tex]
[tex]\[
(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\][/tex]
Coefficients:
[tex]\[ [1, 3, 3, 1] \][/tex]
