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NCERT Class 10 Mathematics Chapter 2 Polynomial
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Polynomial
Chapter โ 2
Exercise 2.1 |
The graphs of v = p(x) are given in fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
Ans: (i) No zeroes.
(ii) 1
(iii) 3
(iv) 2
(v) 4
(vi) 3
Exercise 2.2 |
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
(i) Xยฒ โ 2x โ 8
Ans: xยฒ-2x-8=xยฒ-4x+2x-8=x(x-4)+2(x-4) =(x+2)(x-4)
Zeroes are -2 and 4.
Sum of the zeroes = (-2)+(4) = 2 =
-(-2)/1 = -(Coefficient of x) / (Coefficient of xยฒ
Product of the zeroes = ( โ 2) (4) = (- 8)/1 = (Coefficient term)/Coefficient of xยฒ.
(ii) 4sยฒ โ 4 s +1
Ans: 4sยฒ โ 4 s +1 = ( 2s โ 1)ยฒ
โด The two zeroes are ยฝ, ยฝ
Sum of the zeroes = ยฝ + ยฝ = -(-4)/4 = (Coefficient of ร) /(Coefficient of รยฒ)
Product of two zeroes = (ยฝ) (โ ) =ยผ (Coefficient)/(coefficient of รยฒ) .l
(iii) 6รยฒ-3-7x = 6xยฒ-7x-3 = 6x-9x + 2x-3 = (2x-3) (3x + 1) Putting 2x-3=0 and 3x + 1 = 0
Ans: We get x = 3/2 and x= -1)2 Zeroes of the quadratic polynomial 2 p(x) = 6xยฒ-7x-3
Sum of the two zeroes = (3/2) + (โ ) = 3/2 -โ =7/6
= -(-7)/6 = (Coefficient of ร)/ (coefficient of รยฒ)
Product of the two zeroes = (3/2) ร -(โ ) = -ยฝ = (-3/6)
=(Constant term) / (coefficient of รยฒ)
(iv) 4uยฒ + 8u = 4u (u + 2)
Ans: It gives the two zeroes, 0 and 2 of the polynomial p(u) = 4uยฒ + 8u + u
Sum of the two zeroes 0 + (-) = -2 =(8)/4 = =(Constant term) / (coefficient of รยฒ)
Product of the two zeroes = (0) (-2) = 0 = (Constant term) / (coefficient of รยฒ)
(v) tยฒ-15 = tยฒ (โ15) =( tยฒโ15) (t-โ15)
Ans: If gives the two zeroes of the polynomial p(t) = tยฒ +0t -15 are โ15 and โ15.
-(Coefficient of t)/Coefficient of tยฒ)
Product of the two zeroes
= (-โ15) ร (โ15) = -15 = (-15) = constant Sum of the two zeroes = Product of the two zeroes (-โ15) + โ15=0= Coefficient of term)/(Coefficient of tยฒ)
(vi) 3xยฒ โ ร โ 4 3xยฒ = โ 4ร + 3ร- 4
Ans: 3xยฒ โ ร โ 4 3xยฒ = โ 4ร + 3ร- 4
= 3x(ร + 1) โ 4 (ร + 1) = (ร + ) ( 3x โ 4)
Putting ร + 1 = 0 and 3ร โ 4 = 0, we get = โ 1 and ร = 4/3
i.e ., the two zeroes of the quadratic polynomial p(ร) = 3xยฒ โ ร โ 4 are โ 1 and 4/3
Sum of the two zeroes = (-1) + (4/3) = โ = (-1) /3 = (Coefficient of ร) (Coefficient of xยฒ)
Product of the two zeroes = (-) ร (constant term)/(coefficient of xยฒ)
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4- 1
Ans: Let the polynomial be a xยฒ + b ร c and itโs Zeroes be a and b
Here,a ร b = ยผ and ab = โ 1
Thus the polynomial formed = xยฒ โ ( sum of zeroes ) x + product of Zeroes.
= xยฒ โ ( ยผ) ร โ 1 = xยฒ โ ร/4 โ 1
The other polynomial are = k ( xยฒ- ร/4-1)
If k = 4, then polynomial is = 4xยฒ-ร-4.
(ii) โ2,โ
Ans: Here a+b = โ2 ab=โ
Thus the polynomial formed = xยฒ( sum of Zeroes) ร + product of zeroes.
= xยฒ โ (โ2) ร + โ or xยฒ โ (โ2) ร + โ
Other polynomial are k (xยฒ โ โ2ร+โ )
If k = 3, then the polynomial is 3xยฒ โ 3โ2 ร + 1.
(iii) 0,โ5
Ans: Here a + b = 0 and a .b = โ5
Thus the polynomial formed formed = xยฒ- ( sum of zeroes) ร + product of Zeroes = xยฒ (0) xยฒ (0)x + (โ5 = x + โ5
(vi) 1,1
Ans: Here a+b = a: b=1 this the polynomial formed
= xยฒ ( sum of Zeroes) ร + product of Zeroes = xยฒ โ (1) ร +1 = xยฒ โ ร + 1
(v) ยผ,ยผ
Ans: Here a+b = -ยผ ab = ยผ
Thus the polynomial formed = xยฒ -(sum of zeroes) ร product of Zeroes.
= xยฒ โ (-ยผ) ร + ยผ = xยฒ -1/4ร + ยผ
The other polynomial are k (xยฒ + ยผ ร+ยผ)
If k = 4 ,then the polynomial is 4xยฒ+ ร + 1
(vi) 4,1
Ans: Here a+b = ,ab = 1
Thus the polynomial formed = xยฒ โ (sum of zeroes) ร + product of zeroes.
= xยฒ โ (4) ร + 1 = xยฒ โ 4 ร + 1

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