If $x_{1}, x_{2}, x_{3}, . . . . . x_{n}$ are the roots of $x^{n} + a x + b = 0$ , then the value of $(x_{1} - x_{2}) (x_{1} - x_{3}) (x_{1} - x_{4}) . . . . . (x_{1} - x_{n})$ , is:

n x_{1}^{n} + a

No worries! We‘ve got your back. Try BYJU‘S free classes today!

n x_{1}^{n - 1} + a

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

n x_{1} + a^{n - 1}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

n x_{1} + a^{n}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $n x_{1}^{n - 1} + a$
$x^{n} + a x + b = (x - x_{1}) (x - x_{2}) (x - x_{3}) (x - x_{4}) . . . . . . . (x - x_{n})$
$\Rightarrow (x - x_{2}) (x - x_{3}) (x - x_{4}) . . . . . . . (x - x_{n}) = \frac{x^{n} + a x + b}{x - x_{1}}$
$\Rightarrow {lim}_{x \to x_{1}} (x - x_{2}) (x - x_{3}) (x - x_{4}) . . . . . . . (x - x_{n})$
$= {lim}_{x \to x_{1}} \frac{x^{n} + a x + b}{x - x_{1}}$
$\Rightarrow (x - x_{2}) (x - x_{3}) (x - x_{4}) . . . . . . . (x - x_{n}) = {lim}_{x \to x_{1}} \frac{n x^{n - 1} + a}{1}$
$= n x {_{1}}^{n - 1} + a$

Suggest Corrections

Similar questions

If $x_{1}, x_{2}, x_{3} \dots x_{n}$ are roots of $x^{n} + a x + b = 0$ , then the value of $(x_{1} - x_{2}) (x_{1} - x_{3}) (x_{1} - x_{4}) \dots (x_{1} - x_{n})$ =

Q. If

x_{1}, x_{2}, x_{3}, . ., x_{n}

are the roots of the equation

x^{n} + a x + b = 0,

then the value of

(x_{1} - x_{2}) (x_{1} - x_{3}) (x_{1} - x_{4}) . . . (x_{1} - x_{n})

is equal to

Q. If

x_{1}, x_{2}, x_{3}, x_{4} . . . . x_{2 n + 1}

are in Arithmetic Progression, then find the value of

[\frac{(x_{2 n + 1} - x_{1})}{(x_{2 n + 1} + x_{1})}] + [\frac{(x_{2 n} - x_{2})}{(x_{2 n} + x_{2})}] . . . . . . . . . . . . . . . . . . . . [\frac{(x_{n + 2 - x_{n}})}{(x_{n + 2} + x_{n})}]

Q. Assertion :(A): If

A > 0

B > 0, A + B = \frac{π}{6}

, then the maximum value of

tan A tan B

7 - 4 \sqrt{2}

Reason: (R): If

x_{1} + x_{2} + x_{3} + . . . . . . . x_{n} = λ

(constant), then value of

x_{1}, x_{2}, x_{3}, . . . . . . . x_{n}

is greatest when

x_{1} = x_{2} = x_{3} = . . . . . . . x_{n}

Q. If

x_{1}, x_{2}, x_{3}, . . . x_{n}

are in A.P. whose common difference is

θ

, then the value of

sin θ (sec x_{1} sec x_{2} + sec x_{2} sec x_{3} + . . . + sec x_{n - 1} sec x_{n})

Join BYJU'S Learning Program

If x1,x2,x3,.....xn are the roots of xn+ax+b=0, then the value of (x1−x2)(x1−x3)(x1−x4).....(x1−xn), is:

The correct option is B nxn−11+axn+ax+b=(x−x1)(x−x2)(x−x3)(x−x4).......(x−xn)⇒(x−x2)(x−x3)(x−x4).......(x−xn)=xn+ax+bx−x1⇒limx→x1(x−x2)(x−x3)(x−x4).......(x−xn)=limx→x1xn+ax+bx−x1⇒(x−x2)(x−x3)(x−x4).......(x−xn)=limx→x1nxn−1+a1=nx1n−1+a

If $x_{1}, x_{2}, x_{3}, . . . . . x_{n}$ are the roots of $x^{n} + a x + b = 0$ , then the value of $(x_{1} - x_{2}) (x_{1} - x_{3}) (x_{1} - x_{4}) . . . . . (x_{1} - x_{n})$ , is: