NAVODAYA VIDYALA SAMITI

NOIDA, UP

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E- CONTENT

MATHEMATICS

CLASS X

CHAPTER 2

POLYNOMIALS

Relationship between zeroes and coefficients.

I. Consider the linear polynomial ax + b, a≠0.

Zero is given by x = -b/a

= -( constant term)

coefficient of x

II. Consider the quadratic polynomial p(x) = ax²+bx + c, a≠0.

Let α and β be the two zeroes of this polynomial

By factor theorem , we get that x- α and x- β are factors of p(x).

Therefore, p(x) = k(x- α) ( x- β)

= k[x²- (α + β)x + α β]

ie, ax²+ bx + c = kx²-k( α + β )x + k α β

Equating the coefficients of like terms , we get

k = a, - k(α + β)= b and k α β = c

ie , a = k, α + β = - b/a and α β= c/a

Summary : For the quadratic polynomial ax²+ bx + c, a≠0.

Sum of the zeroes, α + β = - b/a

Product of zeroes, α β= c/a

ie, x= α and x= β are the zeroes of p(x)

III. Consider the cubic polynomial ax³ + bx² + cx + d, a≠0

Let α , β and γ be the three zeroes of this cubic polynomial, then

α + β + γ = -b/a, α β + β γ + α γ = c/a and α β γ= -d/a

For the linear polynomial ax + b, the only one zero is x = -b/a

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For the quadratic polynomial, ax²+ bx + c , Sum of Zeroes, α + β = -b/a

Product of zeroes, α β = c/a

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For the cubic polynomial, ax³ + bx² + cx + d,

α + β + γ = -b/a, α β + β γ + α γ = c/a and α β γ= -d/a

SUMMARY

Question:- Verify the relationship between zeroes and coefficients of the polynomial x²- 2x -8 .

Factorising the polynomial , we get

x²- 2x -8 = (x – 4) (x + 2) [ Since 2=-4 + 2 & -8= -4X2]

Zeroes are given by x - 4 = 0 and x + 2 = 0.

ie x = 4 and x = -2 are the zeroes.

Let α = 4 and β = -2.

Here a = 1, b = -2 and c = -8.

Sum of the zeroes α + β = 2

-b/a = -(-2)/1 =2

Therefore, α + β = - b/a

Similarly, α β= 4 X -2 = -8

c/a = -8.

Therefore α β= c/a.

Question :- Find a quadratic polynomial with ¼ and 2/3 respectively

as the sum and product of the two zeroes.

Solution:-

Let α and β be the two zeroes of the polynomial p(x) = ax²+ bx + c

Therefore, given that α + β = ¼

α β = 2/3

ie, -b/a = ¼

c/a = 2/3

Making the two rational numbers like, we get

-b/a = ¼ =3/12

c/a = 2/3 =8/12

Comparing, we get, a=12, b= -3 , c= 8

Therefore, the polynomial is 12x²-3x + 8.

Question : -Write the cubic polynomial with the sum of zeroes, sum of the products of two zeroes at a time and the product of the zeroes are respectively 3 , 1 and -1/3

Let α,β and γ be the zeroes of the cubic polynomial, ax³ + bx² + cx + d

By the result we have, α+β +γ = -b/a,

α β + β γ + α γ = c/a and α β γ= -d/a

Given that α+β +γ = 3, α β + β γ + α γ = 1 and α β γ= -1/3

Therefore, we get, -b/a = 3

c/a = 1

-d/a = -1/3

Making the denominators equal, we get , -b/a = 3 = 9/3

c/a = 1 = 3/3

and -d/a = -1/3

Equating , we get a= 3, b= -9 , c= 3 and d= 1 so that

the polynomial is 3x³ -9x² + 3x + 1

PREPARED BY

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JOAN A LUKE

JNV KOLLAM

KERALA HYDERABAD REGION