Question

Find the zeros of polynomial x³-15x²+71x-105 given that the zeros are in a.p

Answer 1

mana ki vo zeroes (a-d),a and (a+d) hai
sum of zeroes = -b/a
a-d+a+a+d = 15
3a =15
a= 5
product of zeroes =-d/a
(a-d)(a+d)a=105
a^2-d^2=105/5
25-d^2=21
4 =d^2
d =2
a-d =5-2=3
a=5
a+d=5+2=7

Answer 2

Hey there

Let x = a -d , y = a , and z = a +d , where x y z are the roots of the polynomial respectively

f ( x ) = x ³ - 15x ² +71x -105

∴ x + y + z = -( -15 ) = 15

⇒ ( a - d ) + a ( a + d ) = 15

⇒ 3a = 15 = 5

and xyz = - ( -105 ) = 105

⇒ ( a-d ) a ( a+d ) = 105

⇒ a ( a² -d² ) = 105 [ since ( a+d) ( a -d ) = a²-d²]

⇒5 [ 5² - d² ] = 105

25 - d² = 21

-d² = 21 -25

-d² = -4 =

d = ± 2

CASE -I :- when a = 5 and d = -2 then

( a - d ) , a , ( a + d )

= 5 + 2 , 5 , 5 -2

= 7 , 5 , 3

CASE II :- when a = 5 and d = 2
then,

5 -2 , 5 , 5+2

=3 , 5 , 7

Hence the three zeroes are 7 , 5 , 3 or 3 ,5 ,7


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