The Binomial Expansion

Edexcel IAL WMA12/01/P2 /Oct 2022/Q2 (Remainder theorem, Binomial, Differentiation)

A curve C has equation y = f(x) where​​ 

fx= 2 – kx5

and k is a constant.​​ 

Given that when f(x)is divided by (4x​​ – 5) the remainder is​​ 24332​​ 

• show that​​ k =25​​ 

(2)​​ 

• Find the first three terms, in ascending powers of x, of the binomial expansion of​​ 

2-25x5

giving each term in simplest form.​​ 

(3)​​ 

Using the solution to part (b) and making your method clear,​​ 

• find the gradient of C at the point where x = 0​​ 

(2)

SOLUTION​​ 

a- Using factor theorem, to find the value of​​ k.

(Consider the last bullet point of the flash card above, so oncomparing​​ 4x-5​​ to​​ ax-b,​​ a=4​​ and​​ b=5, and the remainder is​​ f54. Using the fact that the remainder is​​ 24332, substitute​​ x=54​​ and solve​​ the equation.)​​ 

4x-5=0

x=54

f54=24332

2-k×545=24332

2-k×54= 243325

2-54k=32

2-32=54 k 

12=54 k 

2=5k

k=25

b-​​ Using binomial theorem to find the first three terms of the expression.​​ 

Finding the first three terms separately and will then arrange them in the ascending order of power of x.​​ 

an=5025-25x0  =1 x 32 x 1=32

n1an-1 b=5124-25x-1=8 x 16 x-28 x= -32x

n2an-2b2=5223-25x2=10 x 8 x425x2=645x2

2-25x5=32-32x+645x2+…

c- To find the gradient of C, find the​​ first derivative of the given equation or expression.​​ 

fx=2-25x5=32-32x+645x2+…

fx=32-32x+645x2+….

f'x= -32+1285 x

f'0= -32+12850

f'0=-32