Plus 2 Level - Algebra Coaching Classes & Tuition Centers in Guwahati

Answer:

Let   f1(x) =  x^7 - 13x^6 - 3x^5 +295x^4 +515x^3 - 39x^2 -  513x - 243

By trial and error it  can be seen that f1 (1),f1(-1) and f1(-3) each are zero.

f1(1) = 1^7 - 13*1^6 - 3*1^5 +295*1^4 +515*1^3 - 39*1^2 - 513*1- 243 =

1 - 13 - 3 +295 +515 - 39 - 513 - 243  =0. 

Therefore by Factor theorem , (x-1) is a factor of f1(x)

f1( -1) = (-1)^7 - 13(-1)^6 - 3(-1)^5 + 295(-1)^4 + 515(-1)^3 -39(-1)^2 - 513(-1) - 243 =

               -1 -13 + 3 +295 - 515 -39+ 513 -243 =  0. 

Therefore (x+1) is also a factor of f1(x).

f1(-3) =  (-3)^7 - 13(-3)^6 - 3(-3)5 +295(-3)^4 + 515(-3)^3 - 39(-3)^2 -513(-3) - 243 =

 - 2187 -9477 +729 +23895 - 13905 -351 + 1539 -243 = 0.

 Therefore ( x+3) is also a factor of f1(x).

Now (x-1)(x+1)(x+3) =  x^3 +3x^2 - x -3. Dividing f1(x) with (x-1)(x+1)(x+3),we get

f2(x) =   ( x^7 - 13x^6 - 3x^5 +295x^4 +515x^3 - 39x^2 -  513x - 243)/( x^3 +3x^2 -x -3)=

              x^4 -16x^3 +46x^2 +144x +81

It can be seen that f2(-1) is zero. f2(-1) = (-1)^4 -16(-1)^3 +46(-1)^2 +144(-1) +81=

1+ 16 +46 - 144+81 =0.

Therefore (x+1) is a factor of f2(x). 

Now dividing  (x^4 -16x^3 +46x^2 +144x +81) by (x+1) we get

f3(x) = (x^4 -16x^3 +46x^2 +144x +81.) / (x+1)  =   x^3 -17 x^2 +63x+81.

It can be seen that f3 (-1) is zero. f3(-1) = (-1)^3 - 17(-1)^2 +63(-1) +81 =

- 1 - 17 -63 +81 =0. 

Therefore (x+1) is a factor of f3(x).  

Now dividing f3(x) by (x+1),  we get f4(x) =   x^3 -17 x^2 +63x+81/(x+1) =

x^2 -18x+81 ,which can be factorised as (x-9)(x-9).

Thus f1(x) =  x^7 - 13x^6 - 3x^5 +295x^4 +515x^3 - 39x^2 -  513x - 243=

(x+1)^3( x-9)^2(x+3)(x - 1).

Therefore the 7 real roots are  -1, -1,-1 -3 ,9, 9  and 1.

The speed of both Anna and Elsa relative to each other = 60 + 50 = 110 MPH Total distance covered between them = 385 M Hence, time taken for them to meet each other = relative distance covered /relative speed = 385/110 = 3.5 hrs.

Shifts the graph of f(x) to the left by 7 units on the x-axis.

Let us take an example.

If length is twice its breadth,then length is double of the breadth .Mathematically it can be written as length=2*breadth  

x^2-9=0

x^2-3^2=0

according the formula

a^2-b^2=(a+b)(a-b)

(x+3)(x-3)=0

so

(x+3)=0 or (x-3)=0

x=-3   or x=3

Zeros (3,-3)

x^2+2x-15=0

x^2+5x-3x-15=0

x(x+5)-3(x+5)=0

(x+5)(x-3)=0

(x+5)=0   or (x-3)=0

x=-5     or x=3

zeros (3,-5)

x^2-4x+3= 0

then x^2- 3x-x+3=0

x(x-3)-1(x-3)=0

(x-3)[x-1]=0

(x-3)=0   and (x-1)=0

x=3   and   x=1

so  the number representing are 3 and 1

Answer is in the attachment

It depends not only on time but also on concentration.You must concentrate while you study. This concentration comes from your happiness.so always keep happiness.

Since we have given that product of zeros is equal to 1/7. 

Let consider a general polynomial -- ax2+bx+c=0

the product of zeros of this polynomial = c/a

Now let consider all options one by one. 

option 1 -- a = 1, b = 8, c = 7

c/a = 7/1 = 7

option 2 -- a = 1, b = 7, c = 8

c/a = 8/1 = 8

option 3 -- a = 7, b = 8, c = 1

c/a = 1/7

option 4 -- a = 7, b = 1, c = 8

c/a = 8/7

Hence option (3) is correct answer.