CBSE Class 10 Maths HOTs Polynomials

Question. If two zeros of the polynomial f(x) = x⁴ - 6x³ - 26x² + 138x – 35 are 2± √3.Find the other zeros.
Ans: Let the two zeros are 2 + √3 and 2 - √3
Sum of Zeros = 2 + √3 + 2 - √3
= 4
Product of Zeros = (2+ √3)(2 - √3)
= 4 – 3
= 1
Quadratic polynomial is x² – (sum) x + Product 

Question. Find the Quadratic polynomial whose sum and product of zeros are √2 + 1, 1/√2 + 1 .
Ans: sum = 2 √2
Product = 1
Q.P =
X² – (sum) x + Product
∴ x² – (2 √2 ) x + 1

Question. If α,b are the zeros of the polynomial 2x² – 4x + 5 find the value of a) α² + β² b) (α - β)2.
Ans: p (x) = 2 x² – 4 x + 5
α + β = -b/a = 4/2 = 2
αβ = c/a = 5/2
α² + β² = (α + β)² – 2 α β
Substitute then we get, α 2+ β² = -1
(α - β)² = (α + β)² - 4 α β
Substitute, we get = (α - β)² = - 6

Question. If α,b are the zeros of the polynomial x² + 8x + 6 frame a Quadratic polynomial 

Substitute this sum,
We get = 32/3
Required Q.P. is x² - 32/3 x + 32/3

Question. On dividing the polynomial 4x4 - 5x³ - 39x² - 46x – 2 by the polynomial g(x) the quotient is x² - 3x – 5 and the remainder is -5x + 8.Find the polynomial g(x).
Ans: p(x) = g (x) q (x) + r (x)
g(x) = p(x) - r(x)/q(x)
let p(x) = 4x⁴ – 5x³ – 39x² – 46x – 2
q(x) = x² – 3x – 5 and r (x) = -5x + 8
now p(x) – r(x) = 4x⁴ – 5x³ – 39x² – 41x - 10
when p(x) - r(x)/q(x) = 4x² + 7x +2
∴ g(x) = 4x² + 7x + 2

Question. If the squared difference of the zeros of the quadratic polynomial x² + px + 45 is equal to 144 , find the value of p.
Ans: Let two zeros are a and b where α > β
According given condition
(α - β)² = 144
Let p(x) = x² + px + 45
α + β = -b/a = -p/1 = -p
αβ = c/a = 45/1 = 45
now (a - β)² = 144
(α + β)² – 4 αβ = 144
(-p)² – 4 (45) = 144
Solving this we get p = ± 18

Question. If α,β are the zeros of a Quadratic polynomial such that α + β = 24, α - β = 8. Find a Quadratic polynomial having α and β as its zeros.
Ans: α+β = 24
α - β = 8
-----------
2α = 32
α = 32/2 = 16, ∴ α = 16 
Work the same way to α+β = 24 
So, β = 8 
Q.P is x² – (sum) x + product
= x2 – (16+8) x + 16 x 8
Solve this,
it is k (x² – 24x + 128)

Question. If α & β are the zeroes of the polynomial 2x² - 4x + 5, then find the value of a. α² + β² b. 1/ a + 1/ ß c. (α - β)² d. 1/α² + 1/β² e. α³ + β³ 

Question. Obtain all the zeros of the polynomial p(x) = 3x⁴ - 15x³ + 17x² +5x -6 if two zeroes are -1/√3 and 1/√3.
Ans: 3,2

Question. Give examples of polynomials p(x), g(x), q(x) and r(x) which satisfy the division algorithm.
a. deg p(x) = deg q(x) b. deg q(x) = deg r(x) c. deg q(x) = 0.

Question. If the ratios of the polynomial ax³+3bx²+3cx+d are in AP, Prove that 2b³- 3abc+a²d=0
Ans: Let p(x) = ax³ + 3bx² + 3cx + d and α , β , r are their three Zeros but zero are in AP
let a = m – n , b = m, r = m + n 

Question. Find the number of zeros of the polynomial from the graph given. 

Ans: 1

Question. If one zero of the polynomial 3x² - 8x +2k+1 is seven times the other, find the zeros and the value of k
Ans: k= 2/3

Self Practice

Question. If (n-k) is a factor of the polynomials x²+px+q & x² + m x+n. Prove that k = n + n-q/m-p
Ans: since (n – k) is a factor of x² + px + q
∴ (n – k)² + p(n- k) + q = 0
And (n – k)² + m(n – k) + n = 0
Solve this problem by yourself,
∴ k = n + n-q/m-p 

SELF PRACTICE

Question. If 2, ½ are the zeros of px2+5x+r, prove that p= r.

Question. If m, n are zeroes of ax2-5x+c, find the value of a and c if m + n = m.n=10
Ans: a=1/2 ,c=5)

Question. What must be subtracted from 8x4 + 14x³ – 2x² + 7x –8 so that the resulting polynomial is exactly divisible by 4x²+3x-2.
Ans: 14x – 10

Question. What must be added to the polynomial p(x)= x⁴ + 2x³ – 2x² + x –1 so that the resulting polynomial is exactly divisible by x²+2x-3. 
Ans: x-2

Please refer to link below for CBSE Class 10 Mathematics HOTs Polynomials Set A.

Please refer to attached file for CBSE Class 10 Mathematics HOTs Polynomials

Polynomials

 

1.   The value of quadratic polynomial f (x)=2x²- 3x- 2 at x =-2 is ……

2.   If the product of zeroes of the polynomial ax²–6x-6 is 4, find the value of a.

3.   Find the zeroes of the polynomial x²-1. 

4.   The sum and product of the zeroes of a quadratic polynomial are – 1/2 and –3respectively. What is the quadratic polynomial?

5.   Find the number of zeroes of y=p(x) from the graph

6.   Find the zeroes of the polynomial f(x)=4√3x²+ 5x-2√3

7.   2x²-3√x+5 is a polynomial. True or false. Justify

8.   What is the zeroes of the polynomial ax+b=0,a≠0

9.   Give examples of polynomials f(x), g(x) and r(x) which justify the division algorithm f(x) =g(x) q(x)+r(x) and (i) deg r(x)=0 (ii) deg f(x) =deg g(x)=2 (iii) deg q(x) =deg r(x)=1

10. Write a polynomial whose zeroes are √2 and -√2

 

8.   If the polynomial f(x)=x⁴-6x³+16x²-25x+10 is divided by another polynomial x²-2x+k, the remainder comes out to be x+a, find k and a

9.   Find the polynomial of least degree which should be subtracted from the polynomial x⁴+2x³-4x²+6x-3 so that it is exactly divisible by x²-x+1

10. Divide 3x²-x³-3x+5 by x-1-x2 and verify the division algorithm

 

Polynomials(Answer)

1 mark

 

1)12                     2.) -3/2                3) +1             4) 2x²+x-6

  

5) zeros=3           6) -2/√3,√3/4     7) false             8) –b/a

 

9) f(x)=g(x)Xq(x)+r(x), deg r(x)=0                      10) x²-1 x²-1=(x+1)(x-1)+0

 

2/3 marks

 

1.-5,-2

2. 7,-5

3. –2

4. p2-2q

5. k=6

6. a=1, b=+  2, Not a factor

7. –1,-1

8. k=5,a=-5

9. 2x-2

10. Q-(x-2), R=3

 

More Question 

Polynomials

Key points

1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

 

2. A quadratic polynomial in x with real coefficients is of the form ax² + bx + c , where a, b, c are real numbers with a ≠0.

 

3. The zeroes of a polynomial p(x) are precisely the x - coordinates of the points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero of polynomial p(x) if p(a) = 0.

 

4. A polynomial can have at most the same number of zeros as the degree of polynomial.

 

5. For quadratic polynomial ax2 + bx + c (a ≠ 0) Sum of roots = – b / a

Product of roots = c / a

 

6. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that :-

p(x) = g (x).q(x) + r (x), g(x) ≠ 0

wether r(x) = 0 or degree of r (x) < degree of g(x)

 

Polynomials

Questions

1 mark questions ( * Question are under HOTS) :-

 

Q. 1 Write the degree of the polynomial x² – 2x + x³ – x² + 7 .

 

Ans.1. 3

 

Q. 2 Is x = –1, a zero of the polynomial x² – 2x –1 ?

 

Ans2. No

 

Q. 3 Write the coefficient of x in quadratic polynomial x2 – 5x + 6 .

 

Ans3. –5

 

Q. 4 Write the sum of zeros of quadratic polynomial x² –10x +16 .

 

Ans4. 10

 

Q. 5  Write the value of k for which the sum of zeros of the polynomials x² + kx + 3 is 4.

 

Ans5. –4

 

Q. 6 Write the product of zeroes of the quadratic polynomial 2x² – 5x + 3 .

 

Ans6.3/2

 

Q. 7 Write the quadratic polynomial, the product and the sum of whose zeros are –6 and –1 respectively.

 

Ans7. x² + x – 6

 

Q. 8 If x = –2 is a zero of the polynomial x² – 2x – 8 . Write a factor of the given polynomial.

 

Ans8. x+2

 

Q. 9 Write the zero of the polynomial 2x+3.

 

Ans9.– 3/2

 

Q. 10 The graph of y = f(x) is shown in the figure 1. Write the number of zeroes of f(x).

Ans10. 2

 

Q. 11 Write the value of k for which the product of zeroes of the polynomials 2x² +11x – 2k is 5.

 

Ans11. –5

 

Q. 12  Write the sum of zeroes of quadratic polynomial 2x² – 8 .

 

Ans12. 0

 

2 marks questions ( * Question 21-23 under HOTS) :-

 

Q. 13 Find the zeroes of the polynomial 2x² – 8x + 6 .

 

Ans13. 3 and 1

 

Q. 14 Find the zeroes of the polynomial 2x² – 9 .

 

Ans14. 3 √2 and –3√2

 

Q. 15 If the zeroes of a polynomial are –2 and 3, find the polynomial.

 

Ans15. x² – x – 6

 

Q. 16 Find the polynomial whose zeroes are 1/2 and –3/7 .

 

Ans.16. 14x² – x – 3

 

Q. 17 Find the remainder when polynomial 3x² – x³ – 3x + 5 is divided by the polynomial x – x² –1.

 

Ans17. 3

 

Q. 18 Find the polynomial whose zeroes are 3+√ 2 and 3 –√ 2 .

 

Ans.18. x² – 6x + 5

 

Q. 19 Find the polynomial whose zeroes are √5 and – √5.

 

Ans.19. x² – 5

 

Q. 20 Find the zeroes of the polynomial 25x² –15x + 2 .

 

Ans.20. 1/5 and 2/5

 

Q. 21  Find the zeroes of the quadratic polynomial x² + 4 √2x + 6 .

 

Ans.21. – √2 and –3√2

 

Q. 22  Find the zeroes of the quadratic polynomial x² +6 √6x+48 .

 

22. –2 √6 and –4 √6

 

Q. 23  Find the quadratic polynomial whose zeroes are 5 √3/2 and 1/3√ 3.

 

23. 6 √3x² – 47x + 5√ 3

 

3 marks questions ( * Question are under HOTS) :-

 

Q. 24 Using division algorithm check whether the polynomial g (x) = x² + x + 3, is a factor of the polynomial p(x) = x⁴ + x³ – 2x2 – 5x –12.

 

Ans.24. Yes

 

Q. 25 Using division algorithm, find quotient q(x) and remainder r(x) if f(x) is divided by g(x). f (x) = x³ – x² + 4x – 8 ; g (x) = x + 3

 

Ans.25. q(x) = x² – 4x +16 , r (x) = –56

 

Q. 26  If (x+a), is a factor of the polynomials x² + lx + m and x² + nx + k then prove that a = m - k/l - n

 

Q. 27 Find all the zeroes of the polynomial 3x⁴ –15x³ +17x² + 5x – 6 if two zeroes of this polynomial are 1/√3 and– 1/√ 3 .

 

Ans.27. 1/√3 , –1/√3 , 2 and 3

 

Q. 28 On dividing 2x³ + 4x² + 5x + 7 by a polynomial g(x), the quotient and remainder are 2x and 7– 5x respectively, find g(x).

 

Ans.28 x² + 2x + 5

 

Q. 29  If polynomial x⁴ + x³ + 6x² + ax + b is exactly divisible by another polynomial x² +1, find the value of a and b.

 

29. a = 1, b = 5

 

Q. 30 Find all the zeroes of the polynomial x⁴ + x³ – 7x² – 5x +10 if two of its zeroes are 5 and – √ 5.

 

30. x = –2, 1, –√ 5 and √5

 

Q. 31 Find the zeroes of the polynomial 3x² – 27 and verify the relationship between the zeroes and the co-efficients.

 

Ans.31. 3 and –3

 

Q. 32 Find the zeroes of the polynomial 7x² + 2 √14x + 2 and verify the relationship between the zeroes and the co-efficients

 

Ans32. – 2√7 and – √2/7

 

ADDITIONAL QUESTION