Use the binomial theorem to expand the following binomial expressions.
a. (x + y)4
b. (x + 2y)4
c. (x + 2x)4
d. (x − y)4
e. (x − 2xy)4

Step-by-step explanation:

a.

           1

         1  2  1

      1  3  3   1

    1  4    6    4     1

(x+y)^4=1 x^4+4x³y+6 x²y²+4xy3 +1y^4

=x^4+4x³y+6x²y2+4xy³+y^4

or

(x+y)^4=4c0 x^4+4c1 x³y+4c2x²y²+4c3xy³+4c4y^4

4c0=4c4=1

4c1=4/1=4

4c2=(4×3)/(2×1)=6

4c3=4c1=4

so \[(x+a)^4=x^4+4x³y+6x²y²+4xy³+y^4\]

b.

\[(x+2y)^4=x^4+4x³(2y)+6x²(2y)^2+4x(2y)³+(2y)^4\]

=x^4+8x³y+24x²y²+32xy³+16y^4

c.

(x+2x)^4=(3x)^4=81x^4

or(x+2x)^4=x^4+4x³(2x)+6x²(2x)²+4x(2x)³+(2x)^4

=x^4+8x^4+24x^4+32x^4+16x^4

=81x^4

d.

(x-y)^4=x^4+4x³(-y)+6x²(-y)²+4x(-y)³+(-y)^4

=x^4-4x³y+6x²y²-4xy³+y^4

e.

(x-2xy)^4=x^4+4x³(-2xy)+6x²(-2xy)²+4x(-2xy)³+(-2xy)^4

=x^4-8x^4y+24x^4y²-32x^4y³+16x^4y^4

=x^4(1-8y+24y²-32y3+16y^4)

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