MAT1581 ASSIGNMENT 1 SEM 2 2023

MAT1581
ASSIGNMENT 1
SEMESTER 2
2023

, QUESTION 1




Solution:



Binomial theorem:
n
n
(x + y) = ∑ ( ) x n−k y k
n
k
k=0



4
(2x − 3y)4 = (2x + (−3y))
4
4
= ∑ ( ) (2x)4−k (−3y)k
k
k=0

4 4 4
= ( ) (2x)4−0 (−3y)0 + ( ) (2x)4−1 (−3y)1 + ( ) (2x)4−2 (−3y)2
0 1 2
4 4
+ ( ) (2x)4−3 (−3y)3 + ( ) (2x)4−4 (−3y)4
3 4
= 1(2x)4 (1) + 4(2x)3 (−3y)1 + 6(2x)2 (−3y)2 + 4(2x)1 (−3y)3 + 1(2x)0 (−3y)4

= 16x 4 + 4(8x 3 )(−3y) + 6(4x 2 )(9y2 ) + 4(2x)(−27y 3 ) + 1(1)(81y 4 )

= 16x 4 − 96x 3 y + 216x 2 y 2 − 216xy 3 + 81y 4




QUESTION 2




Solution:



x 2 − x − 13 Ax + B C
2
= 2 +
(x + 7)(x − 2) (x + 7) (x − 2)
x 2 − x − 13 (Ax + B)(x − 2) + C(x 2 + 7)
=
(x 2 + 7)(x − 2) (x 2 + 7)(x − 2)