Solutions

Problem Set 4

Algebra

in each case, perform the expansion, and put your answer in an organized form leading from highest exponent of x to lowest.

1. Please expand and simplify

a) (x+y)⁷ x⁷ + 7x⁶y + 21 x⁵y² +35 x⁴y³ + 35x³y⁴ +21 x²y⁵ + 7xy⁶ + y⁷

b) (x – 3y)⁵ x⁵ -15x⁴y +90 x³y² -270x²y³ +405xy⁴ -243y⁵

c) (y + 2x)⁴ y⁴ + 8 y³x +24 y²x² +32 yx³ + 16x⁴

d) (-x – y)⁴ = ( (-1)(x + y ) )⁴ = (-1)⁴ (x+y)⁴ = (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

e) (a +5b)⁵ = a⁵ + 25a⁴ b +250a³b² + 1250a²b³ + 3125ab⁴ + 3125b⁵

2. What is the coefficient of x¹³y⁵ in the expansion of (x+y)²⁰? (think!) 0 since all terms have the total of their exponents being 20, there is no x¹³y⁵ term (which has exponents totalling 18)

3. Please simplify each of the following as much as possible.

a) [ (x + h)³ - x³]/h 3x² + 3xh + h² b) ( (x+ a)⁴ – x⁴ )/a 4x³ +6x²a + 4xa² + a³