Class 10 Polynomials Worksheet

Class 10 Polynomials Worksheet

Class 10 Mathematics Worksheet

Chapter: Polynomials

Name: _______________________ Roll No.: _____________ Date: _____________

Section A – Identify the Degree (10 Questions)

Find the degree of each polynomial.

  1. 5x3−2x+75x^3-2x+75×3−2x+7
  2. 4x5+94x^5+94×5+9
  3. 7x2−5x+37x^2-5x+37×2−5x+3
  4. 8−3x4+x68-3x^4+x^68−3×4+x6
  5. 12x12x12x
  6. 151515
  7. x8+x2+1x^8+x^2+1x8+x2+1
  8. 9x7−4x3+x9x^7-4x^3+x9×7−4×3+x
  9. 2x9−5x6+8x22x^9-5x^6+8x^22×9−5×6+8×2
  10. 11x4−9x2+611x^4-9x^2+611×4−9×2+6

Section B – Find the Zeros (15 Questions)

Find the zero(s) of each polynomial.

  1. x−8x-8x−8
  2. x+15x+15x+15
  3. 2x−102x-102x−10
  4. 5x+205x+205x+20
  5. x2−25x^2-25x2−25
  6. x2−64x^2-64x2−64
  7. x2−16x+64x^2-16x+64x2−16x+64
  8. x2−9x+20x^2-9x+20x2−9x+20
  9. x2−13x+42x^2-13x+42x2−13x+42
  10. x2−11x+24x^2-11x+24x2−11x+24
  11. x2+7x+12x^2+7x+12x2+7x+12
  12. x2−10x+21x^2-10x+21x2−10x+21
  13. 2x2−14x+242x^2-14x+242×2−14x+24
  14. 3x2−12x+93x^2-12x+93×2−12x+9
  15. x2−100x^2-100x2−100

Section C – Verify the Relationship Between Zeros and Coefficients (10 Questions)

Find the zeros and verify:

  1. x2−5x+6x^2-5x+6x2−5x+6
  2. x2−7x+10x^2-7x+10x2−7x+10
  3. x2−8x+15x^2-8x+15x2−8x+15
  4. x2+9x+20x^2+9x+20x2+9x+20
  5. x2−12x+35x^2-12x+35x2−12x+35
  6. 2x2−9x+102x^2-9x+102×2−9x+10
  7. 3x2−13x+123x^2-13x+123×2−13x+12
  8. 4x2−20x+244x^2-20x+244×2−20x+24
  9. 5x2−15x+105x^2-15x+105×2−15x+10
  10. 6x2−11x−106x^2-11x-106×2−11x−10

Section D – Divide Using Polynomial Division (10 Questions)

  1. Divide x2+7x+12x^2+7x+12x2+7x+12 by x+3x+3x+3.
  2. Divide x2−9x^2-9x2−9 by x−3x-3x−3.
  3. Divide 2x2+9x+102x^2+9x+102×2+9x+10 by x+2x+2x+2.
  4. Divide 3x2−15x+183x^2-15x+183×2−15x+18 by x−2x-2x−2.
  5. Divide x3−8x^3-8x3−8 by x−2x-2x−2.
  6. Divide x3+27x^3+27x3+27 by x+3x+3x+3.
  7. Divide 2x3+5x2−4x−32x^3+5x^2-4x-32×3+5×2−4x−3 by x−1x-1x−1.
  8. Divide 3x3−7x2+2x+83x^3-7x^2+2x+83×3−7×2+2x+8 by x+2x+2x+2.
  9. Divide 4x3+8x2−9x−184x^3+8x^2-9x-184×3+8×2−9x−18 by x+2x+2x+2.
  10. Divide 5x3−20x2+15x5x^3-20x^2+15x5×3−20×2+15x by x−1x-1x−1.

Section E – Use the Remainder Theorem (10 Questions)

Find the remainder.

  1. p(x)=2x3+5x2−4x+1,  x−2p(x)=2x^3+5x^2-4x+1,\; x-2p(x)=2×3+5×2−4x+1,x−2
  2. p(x)=x3−7x+9,  x+1p(x)=x^3-7x+9,\; x+1p(x)=x3−7x+9,x+1
  3. p(x)=3x2+2x−5,  x−3p(x)=3x^2+2x-5,\; x-3p(x)=3×2+2x−5,x−3
  4. p(x)=5x3−2x+8,  x+2p(x)=5x^3-2x+8,\; x+2p(x)=5×3−2x+8,x+2
  5. p(x)=4x4−7x+6,  x−1p(x)=4x^4-7x+6,\; x-1p(x)=4×4−7x+6,x−1
  6. p(x)=x4+3x2−8,  x+2p(x)=x^4+3x^2-8,\; x+2p(x)=x4+3×2−8,x+2
  7. p(x)=7x3−5x2+4,  x−4p(x)=7x^3-5x^2+4,\; x-4p(x)=7×3−5×2+4,x−4
  8. p(x)=2x4+3x−5,  x+3p(x)=2x^4+3x-5,\; x+3p(x)=2×4+3x−5,x+3
  9. p(x)=9x2−11x+7,  x−5p(x)=9x^2-11x+7,\; x-5p(x)=9×2−11x+7,x−5
  10. p(x)=6x3+2x2−9,  x+1p(x)=6x^3+2x^2-9,\; x+1p(x)=6×3+2×2−9,x+1

Section F – Use the Factor Theorem (10 Questions)

Determine whether the given polynomial is divisible by the given factor.

  1. x2−5x+6,  x−2x^2-5x+6,\; x-2x2−5x+6,x−2
  2. x2−9,  x+3x^2-9,\; x+3x2−9,x+3
  3. 2x2−8x+8,  x−22x^2-8x+8,\; x-22×2−8x+8,x−2
  4. x3−27,  x−3x^3-27,\; x-3x3−27,x−3
  5. x3+8,  x+2x^3+8,\; x+2x3+8,x+2
  6. 2x3+5x2−3,  x+12x^3+5x^2-3,\; x+12×3+5×2−3,x+1
  7. 4x2−12x+9,  x−324x^2-12x+9,\; x-\frac324×2−12x+9,x−23​
  8. 3x3−24,  x−23x^3-24,\; x-23×3−24,x−2
  9. x2+11x+30,  x+6x^2+11x+30,\; x+6x2+11x+30,x+6
  10. 5x2−20x+15,  x−15x^2-20x+15,\; x-15×2−20x+15,x−1

Section G – Factorise Completely (15 Questions)

  1. x2−49x^2-49x2−49
  2. x2+9x+20x^2+9x+20x2+9x+20
  3. x2−12x+35x^2-12x+35x2−12x+35
  4. 2x2+11x+152x^2+11x+152×2+11x+15
  5. 3x2−15x+183x^2-15x+183×2−15x+18
  6. x3−27x^3-27x3−27
  7. x3+64x^3+64x3+64
  8. x3−8x^3-8x3−8
  9. x3+27x^3+27x3+27
  10. 2x3+10x22x^3+10x^22×3+10×2
  11. 3x2−273x^2-273×2−27
  12. 4x2−254x^2-254×2−25
  13. 5x2−45x5x^2-45x5×2−45x
  14. 6x2+18x6x^2+18x6×2+18x
  15. 9x2−819x^2-819×2−81

Section H – HOTS / Challenge Questions (20 Questions)

  1. Find the quadratic polynomial whose zeros are 5 and 8.
  2. Find the quadratic polynomial whose zeros are -4 and 7.
  3. Find the polynomial whose zeros are 3 and -9.
  4. If one zero of x2−kx+24x^2-kx+24x2−kx+24 is 6, find kkk.
  5. If one zero of x2−11x+mx^2-11x+mx2−11x+m is 3, find mmm.
  6. Find the value of kkk if (x−4)(x-4)(x−4) is a factor of x2+kx−20x^2+kx-20x2+kx−20.
  7. Find kkk if (x+5)(x+5)(x+5) is a factor of x2+kx−30x^2+kx-30x2+kx−30.
  8. If the sum of zeros is 9 and product is 20, form the polynomial.
  9. Form a quadratic polynomial whose zeros are 12\frac1221​ and 3.
  10. Form a polynomial with zeros -2 and -7.
  11. Find all zeros of x3−6x2+11x−6x^3-6x^2+11x-6x3−6×2+11x−6.
  12. Factorise x3+6x2+11x+6x^3+6x^2+11x+6x3+6×2+11x+6.
  13. Find the remainder when x5+2x3+7x^5+2x^3+7x5+2×3+7 is divided by x+1x+1x+1.
  14. Show that (x−5)(x-5)(x−5) is a factor of x3−2x2−5x+50x^3-2x^2-5x+50x3−2×2−5x+50.
  15. If x−3x-3x−3 is a factor of 2x3−kx2−5x+62x^3-kx^2-5x+62×3−kx2−5x+6, find kkk.
  16. Find all factors of x3−9x2+26x−24x^3-9x^2+26x-24x3−9×2+26x−24.
  17. Solve x2−14x+48=0x^2-14x+48=0x2−14x+48=0.
  18. Solve 2x2−18x+40=02x^2-18x+40=02×2−18x+40=0.
  19. Find the polynomial of degree 2 whose sum of zeros is 12 and product is 32.
  20. Create a quadratic polynomial whose zeros differ by 4 and whose sum is 10.

Class 10 Polynomials Worksheet – Answer Key (Part 1)

Questions 1–25

Section A – Degree of the Polynomial

  1. Degree = 3
  2. Degree = 5
  3. Degree = 2
  4. Degree = 6
  5. Degree = 1
  6. Degree = 0
  7. Degree = 8
  8. Degree = 7
  9. Degree = 9
  10. Degree = 4

Section B – Find the Zeros

  1. 8
  2. −15
  3. 5
  4. −4
  5. 5, −5
  6. 8, −8
  7. 8, 8
  8. 4, 5
  9. 6, 7
  10. 3, 8
  11. −3, −4
  12. 3, 7
  13. 3, 4
  14. 1, 3
  15. 10, −10

Class 10 Polynomials Worksheet – Answer Key (Part 2)

Questions 26–50

Section C – Verify the Relationship Between Zeros and Coefficients

For each quadratic (ax^2+bx+c):

  • Sum of zeros = (-\frac{b}{a})
  • Product of zeros = (\frac{c}{a})
  1. (x^2-5x+6)
  • Zeros: 2, 3
  • Sum = 5, Product = 6 ✓
  1. (x^2-7x+10)
  • Zeros: 2, 5
  • Sum = 7, Product = 10 ✓
  1. (x^2-8x+15)
  • Zeros: 3, 5
  • Sum = 8, Product = 15 ✓
  1. (x^2+9x+20)
  • Zeros: −4, −5
  • Sum = −9, Product = 20 ✓
  1. (x^2-12x+35)
  • Zeros: 5, 7
  • Sum = 12, Product = 35 ✓
  1. (2x^2-9x+10)
  • Zeros: 2, 5/2
  • Sum = 9/2, Product = 5 ✓
  1. (3x^2-13x+12)
  • Zeros: 3, 4/3
  • Sum = 13/3, Product = 4 ✓
  1. (4x^2-20x+24)
  • Zeros: 2, 3
  • Sum = 5, Product = 6 ✓
  1. (5x^2-15x+10)
  • Zeros: 1, 2
  • Sum = 3, Product = 2 ✓
  1. (6x^2-11x-10)
  • Zeros: 5/2, −2/3
  • Sum = 11/6, Product = −5/3 ✓

Section D – Polynomial Division

  1. Quotient = x + 4, Remainder = 0
  2. Quotient = x + 3, Remainder = 0
  3. Quotient = 2x + 5, Remainder = 0
  4. Quotient = 3x − 9, Remainder = 0
  5. Quotient = x² + 2x + 4, Remainder = 0
  6. Quotient = x² − 3x + 9, Remainder = 0
  7. Quotient = 2x² + 7x + 3, Remainder = 0
  8. Quotient = 3x² − 13x + 28, Remainder = −48
  9. Quotient = 4x² − 9, Remainder = 0
  10. Quotient = 5x² − 15x, Remainder = 15

Section E – Remainder Theorem

  1. 21
  2. 15
  3. 28
  4. −24
  5. 3

Class 10 Polynomials Worksheet – Answer Key (Part 3)

Questions 51–75

Section E – Remainder Theorem

  1. (p(x)=x^4+3x^2-8,; x+2)
    Remainder = 20
  2. (p(x)=7x^3-5x^2+4,; x-4)
    Remainder = 388
  3. (p(x)=2x^4+3x-5,; x+3)
    Remainder = 148
  4. (p(x)=9x^2-11x+7,; x-5)
    Remainder = 177
  5. (p(x)=6x^3+2x^2-9,; x+1)
    Remainder = −13

Section F – Factor Theorem

Determine whether the given polynomial is divisible by the given factor.

  1. (x^2-5x+6,; x-2) → Yes
  2. (x^2-9,; x+3) → Yes
  3. (2x^2-8x+8,; x-2) → Yes
  4. (x^3-27,; x-3) → Yes
  5. (x^3+8,; x+2) → Yes
  6. (2x^3+5x^2-3,; x+1) → Yes
  7. (4x^2-12x+9,; x-\frac32) → Yes
  8. (3x^3-24,; x-2) → Yes
  9. (x^2+11x+30,; x+6) → Yes
  10. (5x^2-20x+15,; x-1) → Yes

Section G – Factorise Completely

  1. (x^2-49)

= ((x-7)(x+7))

  1. (x^2+9x+20)

= ((x+4)(x+5))

  1. (x^2-12x+35)

= ((x-5)(x-7))

  1. (2x^2+11x+15)

= ((2x+5)(x+3))

  1. (3x^2-15x+18)

= (3(x-2)(x-3))

  1. (x^3-27)

= ((x-3)(x^2+3x+9))

  1. (x^3+64)

= ((x+4)(x^2-4x+16))

  1. (x^3-8)

= ((x-2)(x^2+2x+4))

  1. (x^3+27)

= ((x+3)(x^2-3x+9))

  1. (2x^3+10x^2)

= (2x^2(x+5))

Class 10 Polynomials Worksheet – Answer Key (Part 4)

Questions 76–100

Section G – Factorise Completely

  1. (3x^2-27)

= 3(x-3)(x+3)

  1. (4x^2-25)

= (2x-5)(2x+5)

  1. (5x^2-45x)

= 5x(x-9)

  1. (6x^2+18x)

= 6x(x+3)

  1. (9x^2-81)

= 9(x-3)(x+3)

Section H – HOTS / Challenge Questions

  1. Zeros: 5 and 8

Polynomial: (x^2-13x+40)

  1. Zeros: −4 and 7

Polynomial: (x^2-3x-28)

  1. Zeros: 3 and −9

Polynomial: (x^2+6x-27)

  1. If one zero of (x^2-kx+24) is 6,

Other zero = 4

k = 10

  1. If one zero of (x^2-11x+m) is 3,

Other zero = 8

m = 24

  1. If ((x-4)) is a factor of (x^2+kx-20),

k = 1

  1. If ((x+5)) is a factor of (x^2+kx-30),

k = 1

  1. Sum of zeros = 9, Product = 20

Polynomial: (x^2-9x+20)

  1. Zeros = (\frac12) and 3

Polynomial: (2x^2-7x+3)

  1. Zeros = −2 and −7

Polynomial: (x^2+9x+14)

  1. Find all zeros of

(x^3-6x^2+11x-6)

Zeros: 1, 2, 3

  1. Factorise

(x^3+6x^2+11x+6)

= (x+1)(x+2)(x+3)

  1. Remainder when

(x^5+2x^3+7) is divided by (x+1)

Remainder = 4

  1. Show that ((x-5)) is a factor of

(x^3-2x^2-5x+50)

(p(5)=125-50-25+50=100)

Answer: ((x-5)) is not a factor. (The given question contains an error.)

  1. If ((x-3)) is a factor of

(2x^3-kx^2-5x+6)

(54-9k-15+6=0)

(45-9k=0)

k = 5

  1. Factorise

(x^3-9x^2+26x-24)

= (x-2)(x-3)(x-4)

  1. Solve

(x^2-14x+48=0)

x = 6, 8

  1. Solve

(2x^2-18x+40=0)

(x^2-9x+20=0)

x = 4, 5

  1. Sum of zeros = 12, Product = 32

Polynomial: (x^2-12x+32)

  1. Zeros differ by 4 and sum is 10

Zeros = 3 and 7

Polynomial: (x^2-10x+21)