Let H (x)=3 x4+6 x3-2 x2+1 and g(x) be a polynomial of degree one. If H ((x)/(x-1) x+1)(x-2))=f(x)+(g(x)/(x-1)(x+1)(x-2)) then H (-1)+2 H (2)-3 H (1)=

Question

Let H (x)=3 x4+6 x3-2 x2+1 and g(x) be a polynomial of degree one. If H ((x)/(x-1) x+1)(x-2))=f(x)+(g(x)/(x-1)(x+1)(x-2)) then H (-1)+2 H (2)-3 H (1)= (A) f(-1)+2 f(2)-3 f(1) (B) hH (-1)+f(2)+g(3) (C) g(-1)+2 g(2)-3 g(1) (D) H (1)+2 f(2)-g(1)

Tardigrade Admin · Accepted Answer

We have, (H(x)/(x-1)(x+1)(x-2))=f(x)+(g(x)/(x-1)(x+1)(x-2)) ⇒ H(x)=(x-1)(x+1)(x-2) f(x)+g(x) ⇒ H(-1)=0+g(-1)=g(-1) H(2)=0+g(2)=g(2) H(1)=0+g(1)=g(1) ∴ H(-1)+2 H(2)-3 H(1) =g(-1)+2 g(2)-3 g(1)

Q. Let $H (x) = 3 x^{4} + 6 x^{3} - 2 x^{2} + 1$ and $g (x)$ be a polynomial of degree one. If $H \frac{( x )}{( x - 1 ) x + 1 ) ( x - 2 )} = f (x) + \frac{g ( x )}{( x - 1 ) ( x + 1 ) ( x - 2 )}$ then $H (- 1) + 2 H (2) - 3 H (1) =$

$f (- 1) + 2 f (2) - 3 f (1)$

$h H (- 1) + f (2) + g (3)$

$g (- 1) + 2 g (2) - 3 g (1)$

$H (1) + 2 f (2) - g (1)$

Q. Let H(x)=3x4+6x3−2x2+1 and g(x) be a polynomial of degree one. If H(x−1)x+1)(x−2)(x)​=f(x)+(x−1)(x+1)(x−2)g(x)​ then H(−1)+2H(2)−3H(1)=

f(−1)+2f(2)−3f(1)

hH(−1)+f(2)+g(3)

g(−1)+2g(2)−3g(1)

H(1)+2f(2)−g(1)

We have, (x−1)(x+1)(x−2)H(x)​=f(x)+(x−1)(x+1)(x−2)g(x)​ ⇒H(x)=(x−1)(x+1)(x−2)f(x)+g(x) ⇒H(−1)=0+g(−1)=g(−1) H(2)=0+g(2)=g(2) H(1)=0+g(1)=g(1) ∴H(−1)+2H(2)−3H(1) =g(−1)+2g(2)−3g(1)

Q. Let $H (x) = 3 x^{4} + 6 x^{3} - 2 x^{2} + 1$ and $g (x)$ be a polynomial of degree one. If $H \frac{( x )}{( x - 1 ) x + 1 ) ( x - 2 )} = f (x) + \frac{g ( x )}{( x - 1 ) ( x + 1 ) ( x - 2 )}$ then $H (- 1) + 2 H (2) - 3 H (1) =$

$f (- 1) + 2 f (2) - 3 f (1)$

$h H (- 1) + f (2) + g (3)$

$g (- 1) + 2 g (2) - 3 g (1)$

$H (1) + 2 f (2) - g (1)$