Factor and Remainder Theorems

Edexcel IAL WMA12/01/P2/Oct 2022/Q07 (Indices Differentiation, The Factor Theorem)

The curve C has equation​​ 

y=12x3x-7+14x 13x-1521x               x>0

(a) Write the equation of C in the form​​ 

y = ax72 + bx52 +cx32 + dx12 

 ​​ ​​ ​​ ​​ ​​​​ where a, b, c and d are fully simplified constants.​​ 

(3)​​ 

The curve C has three turning points. Using calculus,​​ 

(b) show that the x coordinates of the three turning points satisfy the equation​​ 

2x3 – 10x2 + 13x – 5 = 0 

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ (3)

Given that the x coordinate of one of the turning points is 1​​ 

(c) find, using algebra, the exact x coordinates of the other two turning points.​​ 

(Solutions based entirely on calculator technology are not acceptable.)

(3)

SOLUTION​​ 

a-​​ 

y=12x4-84x3+182x2-210x21x12

y=12x421x12 -84x321x12+182 x221x12-210x21x12

y=47x4-12-4x3-12+263x2-12-10x12

y=47x72-4x52+263x32-10x12

b-​​ 

(At stationary points, the gradient of the curve is equal to zero. Therefore, we will diffrentiate the equation of curve and equate the diffrentiated expression to zero.)​​ 

y=47x72-4x52+263 x32-10x12

At turning point dydx=0

dydx=72 x47 x52-52x 4x32+32 x263x12-12x 10x-12

dydx=2x52-10x32 +13x12-5x12

2x52-10x32+13x12-5x12=0

2x52-10x32+13x12-5x12=0

2x52+12-10x32+12+13x12+12-5=0

2x3-10x2+13x-5=0

c-​​ Using long division method.

(We can use long division method to divide a polynomial by (x±p), where p is the constant.)

fx=(x-1)(ax2+bx+c)

Where,​​ 

fx=x3+9x2-x-105 

x3+9x2-x-105=x-1(ax2+bx+c)

(ax2+bx+c)=x3+9x2-x-105x-3

                 2x2-3x+5(x-1) 2x3-10x2+13x-5-2x3-2x2            -8x2+13x     -8x2+8x                               5x-5                                5x-5                                0

So, we have

2x3-10x2+3x-5=x-12x2-3x+5

x-12x2-8x+3=0

Now, solving the quadratic equation to find the other two solutions of the turning point.​​ 

 2x2-8x=5=0

2x2-8x= -5

x2-4x= -52

Using the completing square method to solve the quadratic equation.​​ 

x-22-4= -52

x-22= 4-52

x-22=32

x-2=±32

Rationalising the denominator.​​ 

x=2±32×22 

x=2±62