Question

If the remainder when the polynomial f(x) is divided by (x - 1), (x + 1) are 6, 8 respectively, then the remainder when f(x) is divided by (x - 1) (x + 1) is
(1) 7-x
(2) 7+x
(3) 8-x
(4) 8+x
​

Answer 1

Answer:

If the remainder, when polynomial f(x) is divided by x−1,x+1, are 6,8 respectively, then the remainder, when f(x) is divided by(x−1)(x+1) is :

7 - X

Answer 2

Answer:

Remainder is 7 - x

Step-by-step explanation:

Let g(x) = (x - 1)(x + 1)

Since the divisor g(x) is quadratic polynomial, therefore the remainder in general is assumed to be linear. 

Let, remainder =ax + b. 

On Applying Division Algorithm, we get

Dividend = Divisor × Quotient + Remainder

∴ f(x) = Quotient × (x - 1)(x + 1) + ax + b

As polynomial f(x) is divided by (x - 1), (x + 1), remainders are 6, 8 respectively.

So, by remainder theorem, 

∴ f(1) = 6

∴ f(1) = Quotient × (1 - 1)(1 + 1) + a + b

∴ 6 = a + b

∴a + b = 6.......(1)

Also, f(−1) = 8

∴ f(- 1) = Quotient × (- 1 - 1)(- 1 + 1) + a(- 1) + b

 ∴ −a + b = 8......(2)

∴ Adding (1) and (2) equation, we get

$\bf\implies \:2b = 14$

$\bf\implies \:b = 7$

Now, substituting the value of b = 7 in equation (1), we get

$\bf\implies \:a + 7 = 6$

$\bf\implies \:a = - 1$

$\bf\implies \:Remainder \: is \: - x + 7$