Home
Class 12
MATHS
If f(x) is a polynomial of degree <3, pr...

If `f(x)` is a polynomial of degree `<3,` prove that `|1af(a)//(x-a)1bf(b)//(x-b)1cf(c)//(x-c)|-:|1a a^2 1bb^2 1cc^2|=(f(x))/((x-a)(x-b)(x-c))`

Text Solution

Verified by Experts

we have to prove that
`|{:(1,,a,,(f(a))/((x-a))),( 1,,b,,(f(b))/((x-b))),(1,,c,,(f(c))/((x-c))):}| -: |{:(1,,a,,a^(2)),( 1,,b,,b^(2)),(1,,c,,c^(2)):}| =(f(x))/((x-a)(x-b)(x-c))`
`L.H.S.=|{:(1,,a,,(f(a))/((x-a))),( 1,,b,,(f(b))/((x-b))),(1,,c,,(f(c))/((x-c))):}| -: [(a-b)(b-c)(c-a)]`
Expanding along `C_(3)` we get
`L.H.S. =(1)/((a-b)(b-c)(c-a))xx`
`[(f(a)(c-b))/((x-a))+(f(b)(a-c))/((x-b))+(f(c)(b-a))/((x-c))]`
Now using partial fraction method on R.H.S. we get
`R.H.S. (f(x))/((x-a)(x-b)(x-c))=(A)/(x-a)+(B)/(x-b)+(C)/(x-c)` ltbr. `"(As degree of " f(x) lt 3)`
`" then " A= [(f(x))/((x-b)(x-c))]_(x=a)`
`=(f(a))/((b-a)(a-c))`
`" Similarly "=(1)/((b-a)(b-c)) " and " C = (f(c))/((c-a)(c-b))`
`:. R.H.S. =(1)/((a-b)(b-c)(c-a))xx`
`[((c-b)f(a))/((x-b))+((a-c)f(b))/((x-b))+((b-a)f(c))/((x-c))]`
Hence L.H.S. =R.H.S.`
Promotional Banner

Similar Questions

Explore conceptually related problems

If f(x) is a polynomial of nth degree then int e^(x)f(x)dx= Where f^(n)(x) denotes nth order derivative of f(x)w.r.t.x

Let f(x)=(x^(3)+2)^(30) If f^(n)(x) is a polynomial of degree 20 where f^(n)(x) denotes the n^(th) derivativeof f(x) w.r.t x then then value of n is

If int(tan^(9)x)dx=f(x)+log|cosx|, where f(x) is a polynomial of degree n in tan x, then the value of n is

If f (x) is polynomial of degree two and f(0) =4 f'(0) =3,f''(0) =4,then f(-1) =

Evaluate: intf(x)/(x^3-1)dx , where f(x) is a polynomial of degree 2 in x such that f(0)=f(1)=3f(2)=-3

Suppose f(x) is a polynomial of degree four , having critical points at - 1,0,1. If T = (x in R|f (x) = f(0)} , then the sum of sqaure of the elements of T is .

Let f(x) is a polynomial of degree four satisfying f(15)=7 and f(6)=f(8)=f(7)=f(9)=11, then f(1)-f(2)+f(3)-f(4)+.........+f(13)f(14)+f(15) is equal to.

If f(x) is polynomial of degree 5 with leading coefficient =1,f(4)=0. If the curvey y=|f(x)| and y=f(|x|) are same,then find f(5).

Suppose f(x) is a polynomial of degree 5 and with leading coefficient 2009. Suppose further f(1)=1,f(2)=3,f(3)=5,f(4)=7,f(5)=9. What is the value of f(6)