Free · Class 9–10 · 9 worksheets · answer keys · practise online · set 44
Polynomials worksheets — Class 9 and 10, with answers
Free polynomials worksheets for Class 9 and 10: degree, value of a polynomial, zeroes, remainder theorem, multiplying and factorising quadratics, and sum and product of zeroes — with answers.
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The degree is the highest power of x: 4x³ − 2x + 7 has degree 3. Degree 1 = linear, 2 = quadratic, 3 = cubic. Value p(a): put a in place of x. If p(a) = 0, then a is a zero of p(x).Remainder theorem : the remainder when p(x) is divided by (x − a) is p(a). Factor theorem : if p(a) = 0, then (x − a) is a factor.Factorise x² + bx + c: find two numbers that multiply to c and add to b — x² + 5x + 6 = (x + 2)(x + 3). For ax² + bx + c (Class 10): sum of zeroes = −b/a, product = c/a.
Degree of a polynomial (Class 9) Value of a polynomial (Class 9) Zero of a polynomial (Class 9) Remainder theorem (Class 9) Multiply (Class 9) Factorise x² + bx + c (Class 9) Factorise ax² + bx + c (Class 9–10) Zeroes of a quadratic (Class 10) Sum and product of zeroes (Class 10)
Degree of a polynomial Class 9 · Worksheet 1 · set 44
Name: Date: Score: / 10
Write the degree.
Degree of −5x⁶ + 3x⁴ Degree of −5x² − x + 6 Degree of 6x³ + 5x² − 7 Degree of −8x⁵ − 3x³ + x² Degree of 8x⁵ + 7x² − 4 Degree of 9x² + 7x + 7 Degree of −5x − 1 Degree of 8x + 8 Degree of x⁴ − 3x³ − 5x + 7 Degree of −3x⁴ − 6x² + 6x + 6
✅ Answer key — Degree of a polynomial
6 2 3 5 5 2 1 1 4 4 TalentJR · talentjr.in/polynomials-worksheet
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Value of a polynomial Class 9 · Worksheet 2 · set 44
Name: Date: Score: / 8
Find the value.
p(x) = x² − 2x + 7. Find p(3). p(x) = −x² + 5x − 2. Find p(1). p(x) = −4x² − 2x + 7. Find p(−3). p(x) = 4x² + 7x − 7. Find p(0). p(x) = −4x² + 9x + 3. Find p(0). p(x) = 4x² + x − 7. Find p(−2). p(x) = −4x² − 5x + 9. Find p(−2). p(x) = −x² − 5x + 7. Find p(−3).
✅ Answer key — Value of a polynomial
10 2 −23 −7 3 7 3 13 TalentJR · talentjr.in/polynomials-worksheet
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Zero of a polynomial Class 9 · Worksheet 3 · set 44
Name: Date: Score: / 8
Answer Yes or No.
Is x = −4 a zero of p(x) = x² − 9x + 20? Is x = −1 a zero of p(x) = x² + 5x + 4? Is x = −2 a zero of p(x) = x² − 16? Is x = −2 a zero of p(x) = x² + 11x + 30? Is x = 1 a zero of p(x) = x² + x − 2? Is x = 5 a zero of p(x) = x² − 9x + 20? Is x = 6 a zero of p(x) = x² − 4x − 12? Is x = 0 a zero of p(x) = x² + 5x + 4?
✅ Answer key — Zero of a polynomial
No Yes No No Yes Yes Yes No TalentJR · talentjr.in/polynomials-worksheet
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Remainder theorem Class 9 · Worksheet 4 · set 44
Name: Date: Score: / 8
Find the remainder.
Remainder when x³ + 4x² − x − 3 is divided by (x − 3) Remainder when 4x³ − 4x² + 7x − 8 is divided by (x + 2) Remainder when 4x³ − x² − x + 3 is divided by (x + 1) Remainder when 4x³ − 2x² + 3x − 3 is divided by (x − 2) Remainder when 3x³ − 4x² + 8x − 6 is divided by (x + 3) Remainder when 2x³ − 5x² − 4x + 5 is divided by (x + 3) Remainder when 4x³ + x² + 7 is divided by (x − 3) Remainder when 2x³ + 5x² + 2x + 1 is divided by (x − 1)
✅ Answer key — Remainder theorem
57 −70 −1 27 −147 −82 124 10 TalentJR · talentjr.in/polynomials-worksheet
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Multiply Class 9 · Worksheet 5 · set 44
Name: Date: Score: / 10
Multiply using (x + a)(x + b) = x² + (a + b)x + ab.
(x − 2)(x + 2) (x + 9)(x − 6) (x + 5)(x − 3) (x − 8)(x − 8) (x − 1)(x + 7) (x + 8)(x + 9) (x − 9)(x − 8) (x + 2)(x + 5) (x − 5)(x − 5) (x + 2)(x + 2)
✅ Answer key — Multiply
x² − 4 x² + 3x − 54 x² + 2x − 15 x² − 16x + 64 x² + 6x − 7 x² + 17x + 72 x² − 17x + 72 x² + 7x + 10 x² − 10x + 25 x² + 4x + 4 TalentJR · talentjr.in/polynomials-worksheet
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Factorise x² + bx + c Class 9 · Worksheet 6 · set 44
Name: Date: Score: / 10
Factorise.
x² − 12x + 35 x² − 3x − 18 x² − 4x + 3 x² + x − 56 x² + x − 42 x² − 5x − 14 x² + 14x + 48 x² + 11x + 18 x² + 3x − 40 x² + 16x + 63
✅ Answer key — Factorise x² + bx + c
(x − 7)(x − 5) (x − 6)(x + 3) (x − 3)(x − 1) (x − 7)(x + 8) (x − 6)(x + 7) (x − 7)(x + 2) (x + 6)(x + 8) (x + 2)(x + 9) (x − 5)(x + 8) (x + 7)(x + 9) TalentJR · talentjr.in/polynomials-worksheet
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Factorise ax² + bx + c Class 9–10 · Worksheet 7 · set 44
Name: Date: Score: / 8
Factorise by splitting the middle term.
3x² − 14x − 24 3x² + x − 2 3x² + 2x − 5 3x² + 10x + 7 2x² + 3x − 20 3x² − 7x + 2 3x² + 16x − 12 3x² + 10x − 25
✅ Answer key — Factorise ax² + bx + c
(3x + 4)(x − 6) (3x − 2)(x + 1) (3x + 5)(x − 1) (3x + 7)(x + 1) (2x − 5)(x + 4) (3x − 1)(x − 2) (3x − 2)(x + 6) (3x − 5)(x + 5) TalentJR · talentjr.in/polynomials-worksheet
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Zeroes of a quadratic Class 10 · Worksheet 8 · set 44
Name: Date: Score: / 8
Find the zeroes of each polynomial.
x² + 12x + 27 x² + 4x − 45 x² + 2x − 24 x² + 12x + 32 x² + 5x − 6 x² + 6x − 7 x² + 7x + 10 x² − 8x − 9
✅ Answer key — Zeroes of a quadratic
−9 and −3 −9 and 5 −6 and 4 −8 and −4 −6 and 1 −7 and 1 −5 and −2 −1 and 9 TalentJR · talentjr.in/polynomials-worksheet
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Sum and product of zeroes Class 10 · Worksheet 9 · set 44
Name: Date: Score: / 8
Find the sum and the product of the zeroes.
Sum and product of the zeroes of x² + 4x − 11 Sum and product of the zeroes of 5x² + 10x − 5 Sum and product of the zeroes of 5x² + 9x − 10 Sum and product of the zeroes of 2x² + 7x − 6 Sum and product of the zeroes of 2x² + 7x + 1 Sum and product of the zeroes of 3x² + x − 1 Sum and product of the zeroes of x² + 3x − 2 Sum and product of the zeroes of x² − 7x − 10
✅ Answer key — Sum and product of zeroes
sum = −4, product = −11 sum = −2, product = −1 sum = −9/5, product = −2 sum = −7/2, product = −3 sum = −7/2, product = 1/2 sum = −1/3, product = −1/3 sum = −3, product = −2 sum = 7, product = −10 TalentJR · talentjr.in/polynomials-worksheet
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Questions
What is the zero of a polynomial? A value of x that makes the polynomial equal to 0. For x² − 5x + 6 = (x − 2)(x − 3), the zeroes are 2 and 3.
What is the remainder theorem? When p(x) is divided by (x − a), the remainder is p(a) — so you can find the remainder without long division.
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