Free · Class 9–10 · 9 worksheets · answer keys · practise online · set 7
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 7
Name: Date: Score: / 10
Write the degree.
Degree of 2x⁵ − x⁴ + 8x³ + 5x² + 2x Degree of −x² − 4 Degree of −6x⁴ − 6x³ Degree of 6x⁴ + 7x³ Degree of −7x⁵ + 4x Degree of 6x⁴ − 6x³ + 9x² + 6 Degree of 3x − 8 Degree of −2x⁶ + 5x³ + 2x Degree of −7x² − 4x − 4 Degree of 7x³ + 2x² − 9
✅ Answer key — Degree of a polynomial
5 2 4 4 5 4 1 6 2 3 TalentJR · talentjr.in/polynomials-worksheet
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Value of a polynomial Class 9 · Worksheet 2 · set 7
Name: Date: Score: / 8
Find the value.
p(x) = 3x² + 6x − 6. Find p(−2). p(x) = 3x² + 6x − 3. Find p(−1). p(x) = 5x² − 3x + 4. Find p(−2). p(x) = −4x² − 8x + 9. Find p(2). p(x) = −2x² + x + 5. Find p(−3). p(x) = −5x² + 8x + 5. Find p(−1). p(x) = −2x² + 4x − 7. Find p(2). p(x) = 3x² + 7x − 9. Find p(−3).
✅ Answer key — Value of a polynomial
−6 −6 30 −23 −16 −8 −7 −3 TalentJR · talentjr.in/polynomials-worksheet
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Zero of a polynomial Class 9 · Worksheet 3 · set 7
Name: Date: Score: / 8
Answer Yes or No.
Is x = −3 a zero of p(x) = x² − 3x − 18? Is x = −1 a zero of p(x) = x² − 5x − 6? Is x = −4 a zero of p(x) = x² + 5x + 4? Is x = 1 a zero of p(x) = x² − 7x + 6? Is x = −4 a zero of p(x) = x² + 10x + 24? Is x = 6 a zero of p(x) = x² − 3x + 2? Is x = 6 a zero of p(x) = x² − 9x + 18? Is x = 1 a zero of p(x) = x² − 9x + 20?
✅ Answer key — Zero of a polynomial
Yes Yes Yes Yes Yes No Yes No TalentJR · talentjr.in/polynomials-worksheet
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Remainder theorem Class 9 · Worksheet 4 · set 7
Name: Date: Score: / 8
Find the remainder.
Remainder when 3x³ − 2x² + 6x − 8 is divided by (x − 3) Remainder when 4x³ + 4x² − 5x − 5 is divided by (x + 2) Remainder when 3x³ + 3x² + 4x + 9 is divided by (x − 2) Remainder when 4x³ − 3x² + 6x + 2 is divided by (x + 3) Remainder when 4x³ − 5x² − 7x + 7 is divided by (x − 1) Remainder when 4x³ − 3x² − 8x − 2 is divided by (x + 1) Remainder when x³ + 5x² − 9x − 6 is divided by (x − 3) Remainder when 2x³ − 4x² − x + 3 is divided by (x − 1)
✅ Answer key — Remainder theorem
73 −11 53 −151 −1 −1 39 0 TalentJR · talentjr.in/polynomials-worksheet
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Multiply Class 9 · Worksheet 5 · set 7
Name: Date: Score: / 10
Multiply using (x + a)(x + b) = x² + (a + b)x + ab.
(x + 5)(x + 3) (x + 5)(x − 7) (x + 4)(x + 1) (x − 2)(x + 6) (x + 4)(x − 3) (x − 6)(x + 9) (x − 2)(x + 3) (x + 8)(x + 1) (x − 8)(x + 5) (x + 9)(x − 9)
✅ Answer key — Multiply
x² + 8x + 15 x² − 2x − 35 x² + 5x + 4 x² + 4x − 12 x² + x − 12 x² + 3x − 54 x² + x − 6 x² + 9x + 8 x² − 3x − 40 x² − 81 TalentJR · talentjr.in/polynomials-worksheet
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Factorise x² + bx + c Class 9 · Worksheet 6 · set 7
Name: Date: Score: / 10
Factorise.
x² + 3x − 10 x² − 11x + 24 x² + 7x − 8 x² − 11x + 18 x² + x − 20 x² + 11x + 30 x² + 14x + 45 x² + 18x + 81 x² + 9x + 20 x² − 10x + 21
✅ Answer key — Factorise x² + bx + c
(x − 2)(x + 5) (x − 8)(x − 3) (x − 1)(x + 8) (x − 9)(x − 2) (x − 4)(x + 5) (x + 5)(x + 6) (x + 5)(x + 9) (x + 9)(x + 9) (x + 4)(x + 5) (x − 7)(x − 3) TalentJR · talentjr.in/polynomials-worksheet
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Factorise ax² + bx + c Class 9–10 · Worksheet 7 · set 7
Name: Date: Score: / 8
Factorise by splitting the middle term.
4x² + 13x + 10 3x² − 4x − 15 2x² + 11x − 6 2x² + 5x − 3 4x² + 21x − 49 3x² − 19x + 20 4x² − 11x − 3 5x² − 26x + 24
✅ Answer key — Factorise ax² + bx + c
(4x + 5)(x + 2) (3x + 5)(x − 3) (2x − 1)(x + 6) (2x − 1)(x + 3) (4x − 7)(x + 7) (3x − 4)(x − 5) (4x + 1)(x − 3) (5x − 6)(x − 4) TalentJR · talentjr.in/polynomials-worksheet
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Zeroes of a quadratic Class 10 · Worksheet 8 · set 7
Name: Date: Score: / 8
Find the zeroes of each polynomial.
x² − 13x + 42 x² + 15x + 54 x² + x − 2 x² + 5x − 36 x² − 8x − 9 x² + 4x − 45 x² + 9x + 18 x² + 3x − 18
✅ Answer key — Zeroes of a quadratic
6 and 7 −9 and −6 −2 and 1 −9 and 4 −1 and 9 −9 and 5 −6 and −3 −6 and 3 TalentJR · talentjr.in/polynomials-worksheet
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Sum and product of zeroes Class 10 · Worksheet 9 · set 7
Name: Date: Score: / 8
Find the sum and the product of the zeroes.
Sum and product of the zeroes of 3x² − 12x − 3 Sum and product of the zeroes of x² + 11x − 5 Sum and product of the zeroes of 2x² + 5x − 11 Sum and product of the zeroes of x² + 3x + 12 Sum and product of the zeroes of x² − 7x − 7 Sum and product of the zeroes of 5x² − 12x + 6 Sum and product of the zeroes of 3x² − 11x − 12 Sum and product of the zeroes of x² + 8x − 5
✅ Answer key — Sum and product of zeroes
sum = 4, product = −1 sum = −11, product = −5 sum = −5/2, product = −11/2 sum = −3, product = 12 sum = 7, product = −7 sum = 12/5, product = 6/5 sum = 11/3, product = −4 sum = −8, product = −5 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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