Please solve this question..

Question

12x^2 - 7x + 1 using factor theorem; not by middle term splitting

Answer 1

Consider the polynomial function f(x) = 12x² - 7x + 1

The values of x for which f(x)=0 are called the roots of the function. By solving the equation, f(x)=0

Then, we get

12x² - 7x + 1

12x² − 4x − 3x + 1 

= 4x(3x−1)−1(3x−1) 

=(4x−1)(3x−1)

(4x-1) = 0 or (3x-1) = 0

4x = 1 or 3x = 1

x = 1/4 or x = 1/3

Because (4x - 1) and (3x - 1) is a factor of 12x² - 7x + 1 , 1/4 and 1/3 are the solutions to the equation 12x² - 7x + 1 = 0, we can also check as follows:

If x = 1/4 is the solution , then

f(x)= 12x² - 7x + 1

f(1/4) = 12(1/4)² - 7(1/4) + 1

f(1/4) = 12 x 1/16 - 7/4 + 1

f(1/4) = 12/16 - 7/4 + 1

f(1/4) = (28-28)/16

f(1/4)=  0

If x = 1/3 is the solution, then;

f(x)= 12x² - 7x + 1

f(1/3)= 12 (1/3)² - 7(1/3) + 1

f(1/3) = 12 x 1/9 - 7/3 + 1

f(1/3) = 12/9 -7/3 + 1

f(1/3) = (21-21)/9

f(1/3)= 0

If the remainder is zero, (x-c) is a polynomial of f(x).

Answer 2

12x²−7x+1

=12x²−4x−3x+1

=4x(3x−1)−1(3x−1)

=(4x−1)(3x−1)

Comments

This is not factor theorem. You have solved this question using Middle Term splitting.