Polynomials Class 9 Maths Worksheet with Answers Chapter 2

Presenting Class 9 Maths Worksheet with Answers from Chapter 2. In this worksheet, you will find 4 sections – Basic, Standard, Advance, and HOTS. Attempt the worksheet questions only after you have completed the entire chapter. A few of the questions are challenging, not to worry if you are not able to solve them. Try your best to solve the questions below first, then look at the answers and hints for solving them at the end. Here are the major sections covered in the chapter –

• Polynomial basics
• Degree and terms
• Zeroes and values
• Factor and remainder
• Factorisation by splitting

Class 9 Maths Worksheet – Chapter 2: Polynomials

Basic

1. Identify whether each expression is a polynomial in one variable. If yes, write the variable.
   • 4x² − 3x + 7
   • y² + 2
   • 3t + √2
   • y + 2/y
   • x¹⁰ + y³ + t⁵⁰
   • x + √x
2. For p(x) = −x³ + 4x² + 7x − 2, write:
   • Terms
   • Coefficient of x³, x², x
   • Constant term
   • Degree
3. Classify each polynomial as monomial / binomial / trinomial:
   • 7x³
   • x² − 9
   • 3y² + 5y + 7
   • 2 − u − u²
   • 5
   • 0
4. Find the value of each polynomial:
   • p(x) = 5x² − 3x + 7 at x = 1
   • q(y) = 3y³ − 4y + 11 at y = 2
   • r(t) = 4t⁴ + 5t³ − t² + 6 at t = 0
5. Check whether the given numbers are zeroes:
   • p(x) = x + 2 : x = −2, 2
   • q(x) = x² − 2x : x = 0, 2
6. Find the zero of each linear polynomial:
   • p(x) = x + 5
   • q(x) = 2x + 1
   • r(x) = 3x − 2
   • s(x) = 7x
   • t(x) = ax (a ≠ 0)
   • u(x) = cx + d (c ≠ 0)

Standard

1. Write the degree of each polynomial:
   • 5x³ + 4x² + 7x
   • 4 − y²
   • 5t − 7
   • 3
   • x⁷ − 4x⁶ + x + 9
   • 0
2. Classify each as linear / quadratic / cubic (in one variable):
   • x² + x
   • x − x³
   • y + y² + 4
   • 1 + x
   • 3t
   • 7x³
3. Find k if (x − 1) is a factor of p(x):
   • p(x) = 4x³ + 3x² − 4x + k
   • p(x) = x² + x + k
   • p(x) = 2x² + kx + 2
4. Use the Factor Theorem to check whether the given is a factor:
   • Is (x + 2) a factor of x³ + 3x² + 5x + 6?
   • Is (x − 3) a factor of x³ − 4x² + x + 6?
5. Factorise by splitting the middle term:
   • 6x² + 17x + 5
   • 12x² − 7x + 1
   • 2x² + 7x + 3
6. Use identities to expand:
   • (x + 3)²
   • (x − 3)(x + 5)
   • (2x − y + z)²

Advance

1. Using the Remainder Theorem, find the remainder when:
   • p(x) = 2x³ + 5x² − x + 4 is divided by (x − 2)
   • q(x) = x³ − 3x + 7 is divided by (x + 1)
2. Factorise completely:
   • x³ − 2x² − x + 2
   • x³ + 13x² + 32x + 20
3. If x + 1 is a factor, which of these are divisible by (x + 1)?
   • x³ + x² + x + 1
   • x⁴ + x³ + x² + x + 1
   • x⁴ + 3x³ + 3x² + x + 1
4. Expand using identities:
   • (3a + 4b)³
   • (5p − 3q)³
5. Evaluate without multiplying directly:
   • 105 × 106
   • (104)³
   • (999)³
6. Factorise using identities:
   • 49a² + 70ab + 25b²
   • 8x³ + 27y³ + 36x²y + 54xy²

HOTS

1. A polynomial p(x) leaves remainder 5 when divided by (x − 2) and remainder 1 when divided by (x + 1).
   
   Find the remainder when p(x) is divided by (x − 2)(x + 1).
2. Prove (algebraically) that:
   
   (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
3. If x + y + z = 0, show that:
   
   x³ + y³ + z³ = 3xyz
4. Find the value of k so that (x − 2) is a factor of:
   
   p(x) = x³ + kx² − 4x − 8
5. For p(x) = x³ − 23x² + 142x − 120:
   • Verify that x = 1 is a zero.
   • Hence factorise p(x) completely.
6. Without full division, decide which must be true (write Yes/No with reason):
   • If p(3) = 0, then (x − 3) is a factor of p(x).
   • If (x + 5) is a factor of p(x), then p(−5) = 0.
   • A non-zero constant polynomial has a zero.

Answer Key

Basic – Answers

1. • 4x² − 3x + 7: Polynomial in x
   • y² + 2: Polynomial in y
   • 3t + √2: Polynomial in t
   • y + 2/y: Not a polynomial (y⁻¹ term)
   • x¹⁰ + y³ + t⁵⁰: Not one-variable polynomial
   • x + √x: Not a polynomial (x¹ᐟ²)

   Hint: Exponents must be whole numbers and only one variable.

2. Terms: −x³, 4x², 7x, −2
   
   Coefficients: x³ → −1, x² → 4, x → 7
   
   Constant term: −2
   
   Degree: 3
   
   Hint: Degree = highest power.
3. • 7x³: Monomial
   • x² − 9: Binomial
   • 3y² + 5y + 7: Trinomial
   • 2 − u − u²: Trinomial
   • 5: Monomial (constant polynomial)
   • 0: Zero polynomial

   Hint: Count the number of terms.

4. • p(1) = 5 − 3 + 7 = 9
   • q(2) = 3·8 − 8 + 11 = 27
   • r(0) = 6

   Hint: Substitute the value carefully.

5. • x + 2: −2 is a zero, 2 is not
   • x² − 2x: 0 and 2 are both zeroes

   Hint: Check p(value) = 0.

6. • x + 5 → x = −5
   • 2x + 1 → x = −1/2
   • 3x − 2 → x = 2/3
   • 7x → x = 0
   • ax → x = 0
   • cx + d → x = −d/c

   Hint: Set polynomial = 0 and solve.

Standard – Answers

1. • 5x³ + 4x² + 7x: degree 3
   • 4 − y²: degree 2
   • 5t − 7: degree 1
   • 3: degree 0
   • x⁷ − 4x⁶ + x + 9: degree 7
   • 0: degree not defined

   Hint: Zero polynomial has no defined degree.

2. • x² + x: Quadratic
   • x − x³: Cubic
   • y + y² + 4: Quadratic
   • 1 + x: Linear
   • 3t: Linear
   • 7x³: Cubic

   Hint: Use highest power.

3. • 4x³ + 3x² − 4x + k, p(1)=0 ⇒ 4+3−4+k=0 ⇒ k=−3
   • x² + x + k, p(1)=0 ⇒ 1+1+k=0 ⇒ k=−2
   • 2x² + kx + 2, p(1)=0 ⇒ 2+k+2=0 ⇒ k=−4

   Hint: If (x−1) is factor, then p(1)=0.

4. • x³ + 3x² + 5x + 6: p(−2)=0 ⇒ (x+2) is a factor
   • x³ − 4x² + x + 6: p(3)=27−36+3+6=0 ⇒ (x−3) is a factor

   Hint: Check p(a)=0 for factor (x−a).

5. • 6x² + 17x + 5 = (3x + 1)(2x + 5)
   • 12x² − 7x + 1 = (3x − 1)(4x − 1)
   • 2x² + 7x + 3 = (2x + 1)(x + 3)

   Hint: Product of split terms = a·c.

6. • (x + 3)² = x² + 6x + 9
   • (x − 3)(x + 5) = x² + 2x − 15
   • (2x − y + z)² = 4x² + y² + z² − 4xy + 4xz − 2yz

   Hint: Use (a+b+c)² identity.

Advance – Answers

1. • Remainder for (x−2): p(2)=2·8+5·4−2+4=16+20−2+4=38
   • Remainder for (x+1): q(−1)=−1−(−3)+7=9

   Hint: Remainder when divided by (x−a) is p(a).

2. • x³ − 2x² − x + 2 = (x − 2)(x² − 1) = (x − 2)(x − 1)(x + 1)
   • x³ + 13x² + 32x + 20 = (x + 1)(x² + 12x + 20) = (x + 1)(x + 2)(x + 10)

   Hint: Try small integer zeroes first (±1, ±2, ±5…).

3. • x³ + x² + x + 1: Yes (p(−1)=0)
   • x⁴ + x³ + x² + x + 1: No (p(−1)=1)
   • x⁴ + 3x³ + 3x² + x + 1: Yes (p(−1)=0)

   Hint: For factor (x+1), check value at x=−1.

4. • (3a + 4b)³ = 27a³ + 64b³ + 108a²b + 144ab²
   • (5p − 3q)³ = 125p³ − 27q³ − 225p²q + 135pq²

   Hint: Use (x±y)³ identity.

5. • 105×106 = (100+5)(100+6) = 11130
   • (104)³ = (100+4)³ = 1124864
   • (999)³ = (1000−1)³ = 997002999

   Hint: Use (a±b)³ and (a+b)(a+c).

6. • 49a² + 70ab + 25b² = (7a + 5b)²
   • 8x³ + 27y³ + 36x²y + 54xy² = (2x + 3y)³

   Hint: Match with x²+2xy+y² or x³+y³+3xy(x+y).

HOTS – Answers

1. Remainder = 2
   
   Hint: If remainder is R(x)=ax+b, then R(2)=5 and R(−1)=1. Solve: 2a+b=5, −a+b=1.
2. True by expansion:
   
   (x+y+z)² = (x+y)² + 2z(x+y) + z² = x² + y² + z² + 2xy + 2yz + 2zx
   
   Hint: Use (x+y)² first.
3. x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx)
   
   If x+y+z=0 ⇒ x³+y³+z³−3xyz=0 ⇒ x³+y³+z³=3xyz
   
   Hint: Use the identity and substitute.
4. k = 0
   
   Hint: If (x−2) is factor, p(2)=0 ⇒ 8 + 4k − 8 − 8 = 0 ⇒ 4k = 0.
5. p(1)=1−23+142−120=0 ⇒ x=1 is a zero
   
   Factorisation: (x − 1)(x − 10)(x − 12)
   
   Hint: Divide by (x−1) or factor the quadratic.
6. • Yes Hint: Factor Theorem (x−a) ⇔ p(a)=0.
   • Yes Hint: Put a = −5.
   • No Hint: Constant c ≠ 0 never becomes 0.

Worksheet for Other Class 9 Maths Chapters

• Number Systems Class 9 Maths Worksheet Chapter 1
• Linear Equations in Two Variables Class 9 Maths Worksheet Chapter 4
• Introduction to Euclid’s Geometry Class 9 Maths Worksheet Chapter 5
• Heron’s Formula Class 9 Maths Worksheet Chapter 10
• Surface Areas and Volumes Class 9 Maths Worksheet Chapter 11
• Statistics Class 9 Maths Worksheet Chapter 12