find the zeros of the polynomial x^3-15x^2+71x-105, given that the zeros are in A. P
Answers
Answer:3,5,7
Step-by-step explanation:
polynomial by substitution method substitute 3 becuase the powers are not equal so we cant substitute 1 or -1 and when we sub 2 or -2 it will not become zero
3\ 1 -15 71 -105
\ 0 3 -36 105
1 -12 35 0
we want to take 0 first and add and resultant should be multiply with the 3 and we will get 3 values then it will be the quadratic polynomial
now
x^2-12x+35
now factorize
factors of 35 are 1,5,7,35
x^2-7x+5x-35=0
taking x common
x(x-7)-5(x-7)
;x=7 or5
therefore the zeroes are 3,,5,7
therefore these are in ap
Step-by-step explanation:
here, p(x) = x^3-15x^2+71x-105
Let the AP of zeroes be =
a-d , a , a+d
here, a is a constant while d is difference.
let us take the three zeroes as : α = a-d and β = a and γ = a+d
Solution : by using our formulas
--> α + β + γ = - b / a
substituting values
a-d + a + a+d = -(-15)/1
a^3 = 15
a=5 ...(i)
-->α * β * γ= -d/a
a-d * a * a+d = -(-105)/1
lets take the a from β to the other side to make the multiplication easier
(a-d) * ( a+d) = 105/a
(from the algebric equation (a+b)(a-b)=a^2-b^2)
a^2 - d^2 = 105/a
substituting the value of a from (i)
5^2 - d^2 = 105/5
25 - d^2 = 21
25 - 21 = d^2
4 = d^2
2 = d (ii)
now from (i) and (ii) we substitute the values of a and d into the AP of zeroes of the polynomial p(x)
a-d , a , a+d
a-d = 5 - 2 = 3
a = 5
a+d = 5 + 2 = 7
Thus our answer,
α = 3
β = 5
γ = 7
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hope this helps:)