The document contains a mathematical analysis focusing on the polynomial 4x^3 + ax^2 - bx - 2, exploring its first and second derivatives to determine values for a and b based on given conditions. The calculations reveal that a = 11 and b = 5, leading to a specific polynomial form. Additionally, it addresses solving inequalities involving logarithmic functions, ultimately determining that the largest integer satisfying the conditions is n = 5.

1. By Alayya M.H and
Jocelyn A.S
Young’s Modulus
AND
POLYNOMIALS

2. Polynomials
Question
The polynomial 4x3+ax2-bx-2, is denoted by p(x). The
results of differentiating p(x) with respect to x is denoted by
p’(x) and the second derivative is p’’(x).
It is given that when p’(x) is divided by (x+2) the
remainder is -1 and when p”(x) is divided by (x+1)
the remainder is -2.
(i) Find the values of a and b
(ii) When a and b have these values, find the
remainder when p(x) is divided by (x+1)

3. How To Solve
(i)?
y= 4x3+ax2-bx-2
dy/dx = 12x2+2ax-b
p’(x)= 12x2+2ax-b
Step 1: Find the first derivative
Step 2: Input the value of x into the
First derivative
x+2=0
X = -2
p’(x) = 12x2+2ax-b
p’(-2) = 12(-2)2 + 2a(-2) -b
-1 = 48 -4a-b
49 =4a +b

4. dy/dx = 12x2+2ax-b
d2y/dx2 = 24x + 2a
p’’(x)= 24x + 2a
Step 3: Find the second
derivative
Step 4: Input the value of x
into the second derivative
X+1 = 0
X = -1
p’’(x)= 24x + 2a
p’’(-1)= 24(-1) + 2a
-2 = -24 +2a
22 = 2a
11 = a

5. Step 5:
Substitute the value of
A into the equation
found in step 2 to find b
A = 11
49 = 4a +b
49 = 4(11) + b
49 = 44 + b
5 = b
Thus, A is 11 and B is 5

6. Since a is 11 and b is 5,
p(x) = 4x3+11x2-5x-2
How To Solve (ii)?
4x3+11x2-5x-2X +1
4x3+4x2
7x2-5x -2
7x2+7x
- 12x-2
-12x-12
10 Remainder
4x2+7x-12

7. Young’s modulus
question
2 Questions:
1. Solve the inequality |3x - 4| < |3x - 6|
2. Hence find the largest integer n satisfying the inequality |3 ln n-4| < |3 ln n- 6 |

8. How to solve number 1?
Step 1: square both sides of the equation
(3x - 4)2 < (3 x - 6)2
Step 2: expand both sides of the
equation
9x2 - 24x + 16 < 9x2 -36x + 36

9. Step 3
Simplify the equation and find the x !
9x2- 9x2 -24 x + 36 x + 16-36 <0
12 x - 20 < 0
12 x < 20
X < 5/3

10. How to solve number 2 ?
Step 1 : turn the
solution into “ln n “ Step 2 : Find the
largest integer
Let ln n = X
From the solution we
get X < 5/3. So
ln n < 5/3
ln n = 5/3
ln n = logen
logen= 5/3
n = e ^ 5/3
e = natural number
n= 5.294

11. Step 3: Round=off
As we know from the initial
equation in slide 7 that it is less
than, then we need to round it off
to 5 (from 5.294). Thus, n=5 . Do
not round off to 6 because this is
less than not more than.

12. 1. Template : Slidesgo.com
2. Problem inspiration (for polynomials): (9709-W 2009-Paper 3/2-Q5)
References

13. Thank you