Free · Class 9–10 · 9 worksheets · answer keys · practise online · set 43
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 43
Name: Date: Score: / 10
Write the degree.
Degree of 9x⁵ − 3x⁴ + 4 Degree of 6x + 6 Degree of −8x + 7 Degree of 7x² + 9x Degree of −6x⁶ + 4x⁴ + 5 Degree of −9x³ − 9 Degree of −5x + 2 Degree of 2x⁴ + 9x³ + 8x Degree of x² + 7x + 5 Degree of 8x³ − 6x − 7
✅ Answer key — Degree of a polynomial
5 1 1 2 6 3 1 4 2 3 TalentJR · talentjr.in/polynomials-worksheet
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Value of a polynomial Class 9 · Worksheet 2 · set 43
Name: Date: Score: / 8
Find the value.
p(x) = −3x² + 3x + 8. Find p(3). p(x) = x² + 6x − 6. Find p(−2). p(x) = −2x² + 6. Find p(−3). p(x) = 5x² + 5x + 7. Find p(0). p(x) = x² + 4x + 8. Find p(3). p(x) = −5x² + 7x − 4. Find p(−2). p(x) = 4x² + x + 5. Find p(0). p(x) = −2x² − 8x − 1. Find p(−2).
✅ Answer key — Value of a polynomial
−10 −14 −12 7 29 −38 5 7 TalentJR · talentjr.in/polynomials-worksheet
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Zero of a polynomial Class 9 · Worksheet 3 · set 43
Name: Date: Score: / 8
Answer Yes or No.
Is x = −4 a zero of p(x) = x² − x − 6? Is x = 6 a zero of p(x) = x² − 10x + 24? Is x = 2 a zero of p(x) = x² + 12x + 36? Is x = −4 a zero of p(x) = x² − 2x − 24? Is x = 6 a zero of p(x) = x² − 8x + 12? Is x = 0 a zero of p(x) = x² + 3x − 4? Is x = −4 a zero of p(x) = x² − 7x + 12? Is x = −3 a zero of p(x) = x² − x − 20?
✅ Answer key — Zero of a polynomial
No Yes No Yes Yes No No No TalentJR · talentjr.in/polynomials-worksheet
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Remainder theorem Class 9 · Worksheet 4 · set 43
Name: Date: Score: / 8
Find the remainder.
Remainder when 4x³ − 5x² − 3x − 8 is divided by (x − 2) Remainder when 2x³ + 6x² + 7x + 8 is divided by (x − 3) Remainder when 4x³ + 2x² − 9x − 2 is divided by (x − 2) Remainder when 2x³ − 3x² + 7x + 1 is divided by (x + 1) Remainder when 3x³ + 9x − 9 is divided by (x − 2) Remainder when x³ − 4x² − 2x − 8 is divided by (x − 2) Remainder when x³ − 4x² − x + 4 is divided by (x + 2) Remainder when 2x³ + 5x² + 7x + 8 is divided by (x + 3)
✅ Answer key — Remainder theorem
−2 137 20 −11 33 −20 −18 −22 TalentJR · talentjr.in/polynomials-worksheet
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Multiply Class 9 · Worksheet 5 · set 43
Name: Date: Score: / 10
Multiply using (x + a)(x + b) = x² + (a + b)x + ab.
(x − 7)(x + 2) (x − 9)(x + 1) (x + 7)(x − 6) (x + 3)(x + 8) (x − 3)(x + 3) (x + 7)(x + 8) (x − 3)(x − 8) (x + 9)(x + 5) (x − 8)(x + 4) (x − 1)(x + 4)
✅ Answer key — Multiply
x² − 5x − 14 x² − 8x − 9 x² + x − 42 x² + 11x + 24 x² − 9 x² + 15x + 56 x² − 11x + 24 x² + 14x + 45 x² − 4x − 32 x² + 3x − 4 TalentJR · talentjr.in/polynomials-worksheet
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Factorise x² + bx + c Class 9 · Worksheet 6 · set 43
Name: Date: Score: / 10
Factorise.
x² − 10x + 25 x² − 7x − 8 x² + 11x + 18 x² + 2x − 35 x² + x − 30 x² − 6x − 27 x² + 18x + 81 x² + 2x − 63 x² + 12x + 32 x² − 4x − 21
✅ Answer key — Factorise x² + bx + c
(x − 5)(x − 5) (x − 8)(x + 1) (x + 2)(x + 9) (x − 5)(x + 7) (x − 5)(x + 6) (x − 9)(x + 3) (x + 9)(x + 9) (x − 7)(x + 9) (x + 4)(x + 8) (x − 7)(x + 3) TalentJR · talentjr.in/polynomials-worksheet
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Factorise ax² + bx + c Class 9–10 · Worksheet 7 · set 43
Name: Date: Score: / 8
Factorise by splitting the middle term.
4x² + 23x − 35 4x² + 21x − 49 5x² − 42x + 49 5x² + 31x − 28 5x² + 37x + 42 3x² − 17x − 6 4x² − 17x − 15 3x² − 2x − 8
✅ Answer key — Factorise ax² + bx + c
(4x − 5)(x + 7) (4x − 7)(x + 7) (5x − 7)(x − 7) (5x − 4)(x + 7) (5x + 7)(x + 6) (3x + 1)(x − 6) (4x + 3)(x − 5) (3x + 4)(x − 2) TalentJR · talentjr.in/polynomials-worksheet
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Zeroes of a quadratic Class 10 · Worksheet 8 · set 43
Name: Date: Score: / 8
Find the zeroes of each polynomial.
x² − x − 2 x² + 8x + 12 x² + 5x − 6 x² + 7x − 18 x² − x − 12 x² − x − 42 x² − 9x + 8 x² + 2x − 48
✅ Answer key — Zeroes of a quadratic
−1 and 2 −6 and −2 −6 and 1 −9 and 2 −3 and 4 −6 and 7 1 and 8 −8 and 6 TalentJR · talentjr.in/polynomials-worksheet
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Sum and product of zeroes Class 10 · Worksheet 9 · set 43
Name: Date: Score: / 8
Find the sum and the product of the zeroes.
Sum and product of the zeroes of 4x² − 8x + 2 Sum and product of the zeroes of 5x² + 8x − 1 Sum and product of the zeroes of x² + 10x + 1 Sum and product of the zeroes of 5x² + 2x + 7 Sum and product of the zeroes of 4x² + 4x − 6 Sum and product of the zeroes of 2x² + 8x − 9 Sum and product of the zeroes of 5x² − 11x − 8 Sum and product of the zeroes of 2x² − 6x + 4
✅ Answer key — Sum and product of zeroes
sum = 2, product = 1/2 sum = −8/5, product = −1/5 sum = −10, product = 1 sum = −2/5, product = 7/5 sum = −1, product = −3/2 sum = −4, product = −9/2 sum = 11/5, product = −8/5 sum = 3, product = 2 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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