The polynomial R ( x ) is the remainder upon dividing x 2007 byx 2-5 x +6. If R (0) can be expressed as a b(ac-bc), find the value of (a+b+c).

Question

The polynomial R ( x ) is the remainder upon dividing x 2007 byx 2-5 x +6. If R (0) can be expressed as a b(ac-bc), find the value of (a+b+c). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/64a7b2afc7871736e1c390b2a94ee8f4-.png" /> text now R (0)= b =? text Put x =2 text in (1) 2 a + b =22007....(2) text put x =3 text in (1) 3 a + b =32007....(3) (3)-(2) gives a=32007-22007 now b .=22007-2 a=22007-2(32007-22007)=22007+2 ⋅ 22007-2 ⋅ 32007) =3 ⋅ 22007-2 ⋅ 32007=6[22006-32006]=2 ⋅ 3[22006-32006] ≡ a b(ac-bc) text hence a=2 ; b=3 ; c=2006 ∴ a+b+c=2+3+2006=2011

Q. The polynomial R(x) is the remainder upon dividing x2007byx2−5x+6. If R(0) can be expressed as ab(ac−bc), find the value of (a+b+c).

now R(0)=b=? Put x=2 in (1) 2a+b=22007....(2) put x=3 in (1) 3a+b=32007....(3) (3)-(2) gives a=32007−22007 now b=22007−2a=22007−2(32007−22007)=22007+2⋅22007−2⋅32007) =3⋅22007−2⋅32007=6[22006−32006]=2⋅3[22006−32006]≡ab(ac−bc) hence a=2;b=3;c=2006 ∴a+b+c=2+3+2006=2011

Q. The polynomial $R (x)$ is the remainder upon dividing $x^{2007} b y x^{2} - 5 x + 6$ . If $R (0)$ can be expressed as $ab (a^{c} - b^{c})$ , find the value of $(a + b + c)$ .