Mathematics · Textbook solutions

Polynomials

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 163 questions

Introduction, Degree and Types

7 q

Solved Examples

Worked · 7
  1. 3.1 SolvedEx.1
    3m3n6+7m2n3−mn3m^3n^6 + 7m^2n^3 - mn is a polynomial in two variables mm and nn. Write the degree of the polynomial.
  2. For the polynomial p(m)=3m5−7m+5m3+2p(m) = 3m^5 - 7m + 5m^3 + 2 :
    3.1 SolvedEx.2(i)
    Write the polynomial in standard form.
  3. 3.1 SolvedEx.2(ii)
    Write it in the index form by considering all the missing terms with coefficient zero.
  4. 3.1 SolvedEx.2(iii)
    Write it in a coefficient form.
  5. 3.1 SolvedEx.2(iv)
    Write the degree of the polynomial.
  6. 3.1 SolvedEx.3
    Write the polynomial x3+3x−5x^3 + 3x - 5 in coefficient form.
  7. 3.1 SolvedEx.4
    (2,−1,0,5,6)(2, -1, 0, 5, 6) is the coefficient form of the polynomial. Represent it in index form.

Practice Set 3.1

40 q
  1. State whether the given algebraic expressions are polynomials ? Justify.
    Ex 3.1 Q1(i)
    y+1yy + \dfrac{1}{y}
  2. Ex 3.1 Q1(ii)
    2−5x2 - 5\sqrt{x}
  3. Ex 3.1 Q1(iii)
    x2+7x+9x^2 + 7x + 9
  4. Ex 3.1 Q1(iv)
    2m−2+7m−52m^{-2} + 7m - 5
  5. Ex 3.1 Q1(v)
    1010
  6. Write the coefficient of m3m^3 in each of the given polynomial.
    Ex 3.1 Q2(i)
    m3m^3
  7. Ex 3.1 Q2(ii)
    −32+m−3 m3\dfrac{-3}{2} + m - \sqrt{3}\,m^3
  8. Ex 3.1 Q2(iii)
    −23m3−5m2+7m−1\dfrac{-2}{3}m^3 - 5m^2 + 7m - 1
  9. Write the polynomial in xx using the given information.
    Ex 3.1 Q3(i)
    Monomial with degree 7
  10. Ex 3.1 Q3(ii)
    Binomial with degree 35
  11. Ex 3.1 Q3(iii)
    Trinomial with degree 8
  12. Write the degree of the given polynomials.
    Ex 3.1 Q4(i)
    5\sqrt{5}
  13. Ex 3.1 Q4(ii)
    x0x^0
  14. Ex 3.1 Q4(iii)
    x2x^2
  15. Ex 3.1 Q4(iv)
    2 m10−7\sqrt{2}\,m^{10} - 7
  16. Ex 3.1 Q4(v)
    2p−72p - \sqrt{7}
  17. Ex 3.1 Q4(vi)
    7y−y3+y57y - y^3 + y^5
  18. Ex 3.1 Q4(vii)
    xyz+xy−zxyz + xy - z
  19. Ex 3.1 Q4(viii)
    m3n7−3m5n+mnm^3n^7 - 3m^5n + mn
  20. Classify the following polynomials as linear, quadratic and cubic polynomial.
    Ex 3.1 Q5(i)
    2x2+3x+12x^2 + 3x + 1
  21. Ex 3.1 Q5(ii)
    5p5p
  22. Ex 3.1 Q5(iii)
    2 y−12\sqrt{2}\,y - \dfrac{1}{2}
  23. Ex 3.1 Q5(iv)
    m3+7m2+52m−7m^3 + 7m^2 + \dfrac{5}{2}m - \sqrt{7}
  24. Ex 3.1 Q5(v)
    a2a^2
  25. Ex 3.1 Q5(vi)
    3r33r^3
  26. Write the following polynomials in standard form.
    Ex 3.1 Q6(i)
    m3+3+5mm^3 + 3 + 5m
  27. Ex 3.1 Q6(ii)
    −7y+y5+3y3−12+2y4−y2-7y + y^5 + 3y^3 - \dfrac{1}{2} + 2y^4 - y^2
  28. Write the following polynomials in coefficient form.
    Ex 3.1 Q7(i)
    x3−2x^3 - 2
  29. Ex 3.1 Q7(ii)
    5y5y
  30. Ex 3.1 Q7(iii)
    2m4−3m2+72m^4 - 3m^2 + 7
  31. Ex 3.1 Q7(iv)
    −23-\dfrac{2}{3}
  32. Write the polynomials in standard form. (Each is given in coefficient form.)
    Ex 3.1 Q8(i)
    (1,2,3)(1, 2, 3)
  33. Ex 3.1 Q8(ii)
    (5,0,0,0,−1)(5, 0, 0, 0, -1)
  34. Ex 3.1 Q8(iii)
    (−2,2,−2,2)(-2, 2, -2, 2)
  35. Write the appropriate polynomials in the boxes, choosing from x+7x + 7, x2x^2, x3+x2+x+5x^3 + x^2 + x + 5, 2x2+5x+102x^2 + 5x + 10, x3+9x^3 + 9, 3x2+5x3x^2 + 5x.
    Ex 3.1 Q9(i)
    Quadratic polynomial
  36. Ex 3.1 Q9(ii)
    Cubic polynomial
  37. Ex 3.1 Q9(iii)
    Linear polynomial
  38. Ex 3.1 Q9(iv)
    Binomial
  39. Ex 3.1 Q9(v)
    Trinomial
  40. Ex 3.1 Q9(vi)
    Monomial

Operations on Polynomials

5 q

Solved Examples

Worked · 5
  1. 3.2 SolvedEx.1
    Subtract : 5a2−2a5a^2 - 2a from 7a2+5a+67a^2 + 5a + 6.
  2. 3.2 SolvedEx.2
    Multiply : −2a×5a2-2a \times 5a^2
  3. 3.2 SolvedEx.3
    Multiply : (m2−5)×(m3+2m−2)(m^2 - 5) \times (m^3 + 2m - 2)
  4. 3.2 SolvedEx.4
    Add : 3m2n+5mn2−7mn3m^2n + 5mn^2 - 7mn and 2m2n−mn2+mn2m^2n - mn^2 + mn.
  5. 3.2 SolvedEx.5
    Divide (2+2x2)÷(x+2)(2 + 2x^2) \div (x + 2) and write the answer in the form 'Dividend = Divisor ×\times Quotient + Remainder'.

Practice Set 3.2

14 q
  1. Use the given letters to write the answer.
    Ex 3.2 Q1(i)
    There are 'aa' trees in the village Lat. If the number of trees increases every year by 'bb', then how many trees will there be after 'xx' years?
  2. Ex 3.2 Q1(ii)
    For the parade there are yy students in each row and xx such row are formed. Then, how many students are there for the parade in all ?
  3. Ex 3.2 Q1(iii)
    The tens and units place of a two digit number is mm and nn respectively. Write the polynomial which represents the two digit number.
  4. Add the given polynomials.
    Ex 3.2 Q2(i)
    x3−2x2−9x^3 - 2x^2 - 9 ; 5x3+2x+95x^3 + 2x + 9
  5. Ex 3.2 Q2(ii)
    −7m4+5m3+2-7m^4 + 5m^3 + \sqrt{2} ; 5m4−3m3+2m2+3m−65m^4 - 3m^3 + 2m^2 + 3m - 6
  6. Ex 3.2 Q2(iii)
    2y2+7y+52y^2 + 7y + 5 ; 3y+93y + 9 ; 3y2−4y−33y^2 - 4y - 3
  7. Subtract the second polynomial from the first.
    Ex 3.2 Q3(i)
    x2−9x+3x^2 - 9x + \sqrt{3} ; −19x+3+7x2-19x + \sqrt{3} + 7x^2
  8. Ex 3.2 Q3(ii)
    2ab2+3a2b−4ab2ab^2 + 3a^2b - 4ab ; 3ab−8ab2+2a2b3ab - 8ab^2 + 2a^2b
  9. Multiply the given polynomials.
    Ex 3.2 Q4(i)
    2x2x ; x2−2x−1x^2 - 2x - 1
  10. Ex 3.2 Q4(ii)
    x5−1x^5 - 1 ; x3+2x2+2x^3 + 2x^2 + 2
  11. Ex 3.2 Q4(iii)
    2y+12y + 1 ; y2−2y3+3yy^2 - 2y^3 + 3y
  12. Divide first polynomial by second polynomial and write the answer in the form 'Dividend = Divisor ×\times Quotient + Remainder'.
    Ex 3.2 Q5(i)
    x3−64x^3 - 64 ; x−4x - 4
  13. Ex 3.2 Q5(ii)
    5x5+4x4−3x3+2x2+25x^5 + 4x^4 - 3x^3 + 2x^2 + 2 ; x2−xx^2 - x
  14. Ex 3.2 Q6
    Write down the information in the form of algebraic expression and simplify. There is a rectangular farm with length (2a2+3b2)(2a^2 + 3b^2) metre and breadth (a2+b2)(a^2 + b^2) metre. The farmer used a square shaped plot of the farm to build a house. The side of the plot was (a2−b2)(a^2 - b^2) metre. What is the area of the remaining part of the farm ?

Division and Synthetic Division

2 q

Solved Examples

Worked · 2
  1. 3.3 SolvedEx.1
    Divide the polynomial (3x3+2x2−1)(3x^3 + 2x^2 - 1) by (x+2)(x + 2), using the method of synthetic division.
  2. 3.3 SolvedEx.2
    Divide (2y4−3y3+5y−4)÷(y−1)(2y^4 - 3y^3 + 5y - 4) \div (y - 1).

Practice Set 3.3

6 q
  1. Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder.
    Ex 3.3 Q1(i)
    (2m2−3m+10)÷(m−5)(2m^2 - 3m + 10) \div (m - 5)
  2. Ex 3.3 Q1(ii)
    (x4+2x3+3x2+4x+5)÷(x+2)(x^4 + 2x^3 + 3x^2 + 4x + 5) \div (x + 2)
  3. Ex 3.3 Q1(iii)
    (y3−216)÷(y−6)(y^3 - 216) \div (y - 6)
  4. Ex 3.3 Q1(iv)
    (2x4+3x3+4x−2x2)÷(x+3)(2x^4 + 3x^3 + 4x - 2x^2) \div (x + 3)
  5. Ex 3.3 Q1(v)
    (x4−3x2−8)÷(x+4)(x^4 - 3x^2 - 8) \div (x + 4)
  6. Ex 3.3 Q1(vi)
    (y3−3y2+5y−1)÷(y−1)(y^3 - 3y^2 + 5y - 1) \div (y - 1)

Value of a Polynomial

4 q

Solved Examples

Worked · 4
  1. 3.4 SolvedEx.1
    Find the value of the polynomial p(x)=2x2−3x+5p(x) = 2x^2 - 3x + 5 for x=2x = 2.
  2. 3.4 SolvedEx.2
    Find the value of p(y)=2y3−2y+7p(y) = 2y^3 - 2y + \sqrt{7} for y=−2y = -2.
  3. 3.4 SolvedEx.3
    If p(x)=2x2−x3+x+2p(x) = 2x^2 - x^3 + x + 2 then find p(0)p(0).
  4. 3.4 SolvedEx.4
    If the value of the polynomial m2−am+7m^2 - am + 7 for m=−1m = -1 is 10, then find the value of aa.

Practice Set 3.4

4 q
  1. Ex 3.4 Q1
    For x=0x = 0 find the value of the polynomial x2−5x+5x^2 - 5x + 5.
  2. Ex 3.4 Q2
    If p(y)=y2−32 y+1p(y) = y^2 - 3\sqrt{2}\,y + 1 then find p(32)p(3\sqrt{2}).
  3. Ex 3.4 Q3
    If p(m)=m3+2m2−m+10p(m) = m^3 + 2m^2 - m + 10 then p(a)+p(−a)=?p(a) + p(-a) = ?
  4. Ex 3.4 Q4
    If p(y)=2y3−6y2−5y+7p(y) = 2y^3 - 6y^2 - 5y + 7 then find p(2)p(2).

Remainder and Factor Theorems

8 q

Solved Examples

Worked · 8
  1. 3.5 SolvedEx.1
    Divide p(x)=(4x2−x+2)p(x) = (4x^2 - x + 2) by (x+1)(x + 1), and verify that the remainder equals the value of p(x)p(x) at x=−1x = -1.
  2. 3.5 SolvedEx.2
    Divide x4−5x2−4xx^4 - 5x^2 - 4x by x+3x + 3 and find the remainder.
  3. 3.5 SolvedEx.3
    By using remainder theorem divide the polynomial x3−2x2−4x−1x^3 - 2x^2 - 4x - 1 by x−1x - 1 and find the remainder.
  4. 3.5 SolvedEx.4
    If the polynomial t3−3t2+kt+50t^3 - 3t^2 + kt + 50 is divided by (t−3)(t - 3), the remainder is 62. Find the value of kk.
  5. 3.5 SolvedEx.5
    If p(x)=(x3+4x−5)p(x) = (x^3 + 4x - 5) is divided by (x−1)(x - 1) then find the remainder and hence determine whether (x−1)(x - 1) is a factor of p(x)p(x) or not ?
  6. 3.5 SolvedEx.6
    If p(x)=x3+4x−5p(x) = x^3 + 4x - 5 is divided by x+2x + 2 then find the remainder and hence determine whether (x+2)(x + 2) is a factor of p(x)p(x) or not.
  7. 3.5 SolvedEx.7
    Check whether x−2x - 2 is a factor of the polynomial x3−x2−4x^3 - x^2 - 4 by using factor theorem.
  8. 3.5 SolvedEx.8
    If (x−1)(x - 1) is the factor of the polynomial (x3−2x2+mx−4)(x^3 - 2x^2 + mx - 4) then find the value of mm.

Practice Set 3.5

21 q
  1. Find the value of the polynomial 2x−2x3+72x - 2x^3 + 7 using given values for xx.
    Ex 3.5 Q1(i)
    x=3x = 3
  2. Ex 3.5 Q1(ii)
    x=−1x = -1
  3. Ex 3.5 Q1(iii)
    x=0x = 0
  4. For each of the following polynomial, find p(1)p(1), p(0)p(0) and p(−2)p(-2).
    Ex 3.5 Q2(i)
    p(x)=x3p(x) = x^3
  5. Ex 3.5 Q2(ii)
    p(y)=y2−2y+5p(y) = y^2 - 2y + 5
  6. Ex 3.5 Q2(iii)
    p(x)=x4−2x2−xp(x) = x^4 - 2x^2 - x
  7. Ex 3.5 Q3
    If the value of the polynomial m3+2m+am^3 + 2m + a is 12 for m=2m = 2, then find the value of aa.
  8. Ex 3.5 Q4
    For the polynomial mx2−2x+3mx^2 - 2x + 3 if p(−1)=7p(-1) = 7 then find mm.
  9. Divide the first polynomial by the second polynomial and find the remainder using remainder theorem.
    Ex 3.5 Q5(i)
    (x2−7x+9)(x^2 - 7x + 9) ; (x+1)(x + 1)
  10. Ex 3.5 Q5(ii)
    (2x3−2x2+ax−a)(2x^3 - 2x^2 + ax - a) ; (x−a)(x - a)
  11. Ex 3.5 Q5(iii)
    (54m3+18m2−27m+5)(54m^3 + 18m^2 - 27m + 5) ; (m−3)(m - 3)
  12. Ex 3.5 Q6
    If the polynomial y3−5y2+7y+my^3 - 5y^2 + 7y + m is divided by y+2y + 2 and the remainder is 50 then find the value of mm.
  13. Ex 3.5 Q7
    Use factor theorem to determine whether x+3x + 3 is factor of x2+2x−3x^2 + 2x - 3 or not.
  14. Ex 3.5 Q8
    If (x−2)(x - 2) is a factor of x3−mx2+10x−20x^3 - mx^2 + 10x - 20 then find the value of mm.
  15. By using factor theorem in the following examples, determine whether q(x)q(x) is a factor p(x)p(x) or not.
    Ex 3.5 Q9(i)
    p(x)=x3−x2−x−1p(x) = x^3 - x^2 - x - 1, q(x)=x−1q(x) = x - 1
  16. Ex 3.5 Q9(ii)
    p(x)=2x3−x2−45p(x) = 2x^3 - x^2 - 45, q(x)=x−3q(x) = x - 3
  17. Ex 3.5 Q10
    If (x31+31)(x^{31} + 31) is divided by (x+1)(x + 1) then find the remainder.
  18. Ex 3.5 Q11
    Show that m−1m - 1 is a factor of m21−1m^{21} - 1 and m22−1m^{22} - 1.
  19. Ex 3.5 Q12
    If x−2x - 2 and x−12x - \dfrac{1}{2} both are the factors of the polynomial nx2−5x+mnx^2 - 5x + m, then show that m=n=2m = n = 2.
  20. Find the required value of the polynomial.
    Ex 3.5 Q13(i)
    If p(x)=2+5xp(x) = 2 + 5x then p(2)+p(−2)−p(1)p(2) + p(-2) - p(1).
  21. Ex 3.5 Q13(ii)
    If p(x)=2x2−53 x+5p(x) = 2x^2 - 5\sqrt{3}\,x + 5 then p(53)p(5\sqrt{3}).

Factorisation of Polynomials

6 q

Solved Examples

Worked · 6
  1. Factorize.
    3.6 SolvedEx.1
    4x2−254x^2 - 25
  2. 3.6 SolvedEx.2
    3x2+7x+23x^2 + 7x + 2
  3. 3.6 SolvedEx.3
    63x2+5x−263x^2 + 5x - 2
  4. 3.6 SolvedEx.4
    6x2−5x−66x^2 - 5x - 6
  5. 3.6 SolvedEx.5
    Factorise : (y2−3y)2−5(y2−3y)−50(y^2 - 3y)^2 - 5(y^2 - 3y) - 50.
  6. 3.6 SolvedEx.6
    Factorise : (x+2)(x−3)(x−7)(x−2)+64(x + 2)(x - 3)(x - 7)(x - 2) + 64.

Practice Set 3.6

13 q
  1. Find the factors of the polynomials given below.
    Ex 3.6 Q1(i)
    2x2+x−12x^2 + x - 1
  2. Ex 3.6 Q1(ii)
    2m2+5m−32m^2 + 5m - 3
  3. Ex 3.6 Q1(iii)
    12x2+61x+7712x^2 + 61x + 77
  4. Ex 3.6 Q1(iv)
    3y2−2y−13y^2 - 2y - 1
  5. Ex 3.6 Q1(v)
    3 x2+4x+3\sqrt{3}\,x^2 + 4x + \sqrt{3}
  6. Ex 3.6 Q1(vi)
    12x2−3x+4\dfrac{1}{2}x^2 - 3x + 4
  7. Factorize the following polynomials.
    Ex 3.6 Q2(i)
    (x2−x)2−8(x2−x)+12(x^2 - x)^2 - 8(x^2 - x) + 12
  8. Ex 3.6 Q2(ii)
    (x−5)2−(5x−25)−24(x - 5)^2 - (5x - 25) - 24
  9. Ex 3.6 Q2(iii)
    (x2−6x)2−8(x2−6x+8)−64(x^2 - 6x)^2 - 8(x^2 - 6x + 8) - 64
  10. Ex 3.6 Q2(iv)
    (x2−2x+3)(x2−2x+5)−35(x^2 - 2x + 3)(x^2 - 2x + 5) - 35
  11. Ex 3.6 Q2(v)
    (y+2)(y−3)(y+8)(y+3)+56(y + 2)(y - 3)(y + 8)(y + 3) + 56
  12. Ex 3.6 Q2(vi)
    (y2+5y)(y2+5y−2)−24(y^2 + 5y)(y^2 + 5y - 2) - 24
  13. Ex 3.6 Q2(vii)
    (x−3)(x−4)2(x−5)−6(x - 3)(x - 4)^2(x - 5) - 6

Problem Set 3

33 q
  1. Write the correct alternative answer for each of the following questions.
    Prob Q1(i)
    Which of the following is a polynomial ?
    1. A.
      xy\dfrac{x}{y}
    2. B.
      x2−3xx^{\sqrt{2}} - 3x
    3. C.
      x−2+7x^{-2} + 7
    4. D.
      2 x2+12\sqrt{2}\,x^2 + \dfrac{1}{2}
  2. Prob Q1(ii)
    What is the degree of the polynomial 7\sqrt{7} ?
    1. A.
      12\dfrac{1}{2}
    2. B.
      5
    3. C.
      2
    4. D.
      0
  3. Prob Q1(iii)
    What is the degree of the 0 polynomial ?
    1. A.
      0
    2. B.
      1
    3. C.
      undefined
    4. D.
      any real number
  4. Prob Q1(iv)
    What is the degree of the polynomial 2x2+5x3+72x^2 + 5x^3 + 7 ?
    1. A.
      3
    2. B.
      2
    3. C.
      5
    4. D.
      7
  5. Prob Q1(v)
    What is the coefficient form of x3−1x^3 - 1 ?
    1. A.
      (1,−1)(1, -1)
    2. B.
      (3,−1)(3, -1)
    3. C.
      (1,0,0,−1)(1, 0, 0, -1)
    4. D.
      (1,3,−1)(1, 3, -1)
  6. Prob Q1(vi)
    p(x)=x2−77 x+3p(x) = x^2 - 7\sqrt{7}\,x + 3 then p(77)=?p(7\sqrt{7}) = ?
    1. A.
      3
    2. B.
      777\sqrt{7}
    3. C.
      427+342\sqrt{7} + 3
    4. D.
      49749\sqrt{7}
  7. Prob Q1(vii)
    When x=−1x = -1, what is the value of the polynomial 2x3+2x2x^3 + 2x ?
    1. A.
      4
    2. B.
      2
    3. C.
      −2-2
    4. D.
      −4-4
  8. Prob Q1(viii)
    If x−1x - 1 is a factor of the polynomial 3x2+mx3x^2 + mx then find the value of mm.
    1. A.
      2
    2. B.
      −2-2
    3. C.
      −3-3
    4. D.
      3
  9. Prob Q1(ix)
    Multiply (x2−3)(2x−7x3+4)(x^2 - 3)(2x - 7x^3 + 4) and write the degree of the product.
    1. A.
      5
    2. B.
      3
    3. C.
      2
    4. D.
      0
  10. Prob Q1(x)
    Which of the following is a linear polynomial ?
    1. A.
      x+5x + 5
    2. B.
      x2+5x^2 + 5
    3. C.
      x3+5x^3 + 5
    4. D.
      x4+5x^4 + 5
  11. Write the degree of the polynomial for each of the following.
    Prob Q2(i)
    5+3x45 + 3x^4
  12. Prob Q2(ii)
    77
  13. Prob Q2(iii)
    ax7+bx9ax^7 + bx^9 (aa, bb are constants.)
  14. Write the following polynomials in standard form.
    Prob Q3(i)
    4x2+7x4−x3−x+94x^2 + 7x^4 - x^3 - x + 9
  15. Prob Q3(ii)
    p+2p3+10p2+5p4−8p + 2p^3 + 10p^2 + 5p^4 - 8
  16. Write the following polynomial in coefficient form.
    Prob Q4(i)
    x4+16x^4 + 16
  17. Prob Q4(ii)
    m5+2m2+3m+15m^5 + 2m^2 + 3m + 15
  18. Write the index form of the polynomial using variable xx from its coefficient form.
    Prob Q5(i)
    (3,−2,0,7,18)(3, -2, 0, 7, 18)
  19. Prob Q5(ii)
    (6,1,0,7)(6, 1, 0, 7)
  20. Prob Q5(iii)
    (4,5,−3,0)(4, 5, -3, 0)
  21. Add the following polynomials.
    Prob Q6(i)
    7x4−2x3+x+107x^4 - 2x^3 + x + 10 ; 3x4+15x3+9x2−8x+23x^4 + 15x^3 + 9x^2 - 8x + 2
  22. Prob Q6(ii)
    3p3q+2p2q+73p^3q + 2p^2q + 7 ; 2p2q+4pq−2p3q2p^2q + 4pq - 2p^3q
  23. Subtract the second polynomial from the first.
    Prob Q7(i)
    5x2−2y+95x^2 - 2y + 9 ; 3x2+5y−73x^2 + 5y - 7
  24. Prob Q7(ii)
    2x2+3x+52x^2 + 3x + 5 ; x2−2x+3x^2 - 2x + 3
  25. Multiply the following polynomials.
    Prob Q8(i)
    (m3−2m+3)(m4−2m2+3m+2)(m^3 - 2m + 3)(m^4 - 2m^2 + 3m + 2)
  26. Prob Q8(ii)
    (5m3−2)(m2−m+3)(5m^3 - 2)(m^2 - m + 3)
  27. Prob Q9
    Divide polynomial 3x3−8x2+x+73x^3 - 8x^2 + x + 7 by x−3x - 3 using synthetic method and write the quotient and remainder.
  28. Prob Q10
    For which the value of mm, x+3x + 3 is the factor of the polynomial x3−2mx+21x^3 - 2mx + 21 ?
  29. Prob Q11
    At the end of the year 2016, the population of villages Kovad, Varud, Chikhali is 5x2−3y25x^2 - 3y^2, 7y2+2xy7y^2 + 2xy and 9x2+4xy9x^2 + 4xy respectively. At the beginning of the year 2017, x2+xy−y2x^2 + xy - y^2, 5xy5xy and 3x2+xy3x^2 + xy persons from each of the three villages respectively went to another village for education then what is the remaining total population of these three villages ?
  30. Prob Q12
    Polynomials bx2+x+5bx^2 + x + 5 and bx3−2x+5bx^3 - 2x + 5 are divided by polynomial x−3x - 3 and the remainders are mm and nn respectively. If m−n=0m - n = 0 then find the value of bb.
  31. Prob Q13
    Simplify. (8m2+3m−6)−(9m−7)+(3m2−2m+4)(8m^2 + 3m - 6) - (9m - 7) + (3m^2 - 2m + 4)
  32. Prob Q14
    Which polynomial is to be subtracted from x2+13x+7x^2 + 13x + 7 to get the polynomial 3x2+5x−43x^2 + 5x - 4 ?
  33. Prob Q15
    Which polynomial is to be added to 4m+2n+34m + 2n + 3 to get the polynomial 6m+3n+106m + 3n + 10 ?