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# If the Zeros of the Polynomial F(X) = 2x3 − 15x2 + 37x − 30 Are in A.P., Find Them. - CBSE Class 10 - Mathematics

ConceptRelationship Between Zeroes and Coefficients of a Polynomial

#### Question

If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.

#### Solution

Let α = a - d, β = a and γ = a + d be the zeros of the polynomial

f(x) = 2x3 − 15x2 + 37x − 30

Therefore

alpha+beta+gamma=("coefficient of "x^2)/("coefficient of "x^3)

=-((-15)/2)

=15/2

alphabetagamma="-constant term"/("coefficient of "x^2)

=-((-30)/2)

= 15

Sum of the zeroes =("coefficient of "x^2)/("coefficient of "x^3)

(a-d)+a+(a+d)=15/2

a+a+a-d+d=15/2

3a=15/2

a=15/2xx1/3

a=5/2

Product of the zeroes ="-constant term"/("coefficient of "x^2)

alphabetagamma=15

(a-d)+a+(a+d)=15

a(a^2-d^2)=15

Substituting a = 5/2 we get

5/2((5/2)^2-d^2)=15

5/2(25/4-d^2)=15

25/4-d^2=15xx2/5

25/4-d^2=3xx2

25/4-d^2=6

-d^2=6-25/4

-d^2=(24-25)/4

-d^2=(-1)/4

d^2=1/4

d xx d=1/2xx1/2

d=1/2

Therefore, substituting a=5/2 and d=1/2 in α = a - d, β = a and γ = a + d

α = a - d

alpha=5/2-1/2

alpha=(5-1)/2

alpha=4/2

alpha=2

β = a

beta=5/2

γ = a + d

gamma=5/2+1/2

gamma=(5+1)/2

gamma=6/2

gamma=3

Hence, the zeros of the polynomial are 2, 5/2 , 3.

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Solution for 10 Mathematics (2018 to Current)
Chapter 2: Polynomials
Ex. 2.20 | Q: 3 | Page no. 43

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Solution If the Zeros of the Polynomial F(X) = 2x3 − 15x2 + 37x − 30 Are in A.P., Find Them. Concept: Relationship Between Zeroes and Coefficients of a Polynomial.
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