(i) 4x2 - 3x + 5 Solution / : It is a polynomial because the exponent of variable x in all terms is a whole number (2, 1, 0). / x (ii) z + 3/z Solution / : It is not a polynomial because z + 3/z = z + 3z-1, where the power of z is -1, which is not a ...

हल / हल: It is a polynomial because the exponent of variable x in all terms is a whole number (2, 1, 0). / यह एक बहुपद है क्योंकि चर x की घातें पूर्ण संख्याएँ हैं। हल / हल: It is not a polynomial because z + 3/z = z + 3z-1, where the power of z is -1, which is not a whole number. / यह बहुपद नहीं है क्योंकि z + 3z-1 में चर z की घात -1 है जो पूर्ण संख्या नहीं है। हल / हल: It is not a polynomial because √y = y1/2, where the exponent is 1/2, which is not a whole number. / यह बहुपद नहीं है क्योंकि चर y की घात 1/2 है जो पूर्ण संख्या नहीं है। हल / हल: It is a polynomial because the powers of variable...

CG Board Class 9 Mathematics Chapter 4 Solutions: Polynomials (बहुपद) - 2026-27

CG SCERT (Chhattisgarh Board) Class 9 Mathematics Solutions | 2026-27 Official Edition

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End-of-Chapter Exercises (Solutions)

Question 1

(i) 4×2 – 3x + 5 Solution / : It is a polynomial because the…

📝 Question:

(i) 4x² – 3x + 5
Solution / : It is a polynomial because the exponent of variable x in all terms is a whole number (2, 1, 0). / x
(ii) z + 3/z
Solution / : It is not a polynomial because z + 3/z = z + 3z^-1, where the power of z is -1, which is not a whole number. / z + 3z^-1 z -1
(iii) √y + 2y + 3 (or y^1/2 + 2y + 3)
Solution / : It is not a polynomial because √y = y^1/2, where the exponent is 1/2, which is not a whole number. / y 1/2
(iv) x² + 2/3x + 2
Solution / : It is a polynomial because the powers of variable x are whole numbers. / x
(v) x¹⁰ + y³ + t⁵⁰

(i) 3x³ + 2x² + 3x + 2 (Textbook variation 3x² + 2x² + 3x + 2)
Solution / : Combining 3x² + 2x² = 5x², the coefficient of x² is 5 (or 2 if 3x³ term). / x² 5
(ii) 3x² + 1
Solution / : Coefficient of x² is 3. / x² 3
(iii) 2 – 5x² + 1/2x³
Solution / : Coefficient of x² is -5. / x² -5
(iv) 1/2x² + 2
Solution / : Coefficient of x² is 1/2. / x² 1/2
(v) x⁴ + x³ + 1/4x²
Solution / : Coefficient of x² is 1/4. / x² 1/4

(i) x² + 1/5x + 5
Solution / : Coefficient of x = 1/5, Constant term = 5. / x = 1/5, = 5
(ii) x² + 7
Solution / : Coefficient of x = 0, Constant term = 7. / x = 0, = 7
(iii) x² + 2x
Solution / : Coefficient of x = 2, Constant term = 0. / x = 2, = 0

(i) Binomial of degree 4 / 4 :
Solution: x⁴ + 3 or 3x⁴ + 4
(ii) Trinomial of degree 6 / 6 :
Solution: y⁶ + 3y + 3 or t⁶ + 3t – 5
(iii) Monomial of degree 5 / 5 :
Solution: 3x⁵ or z⁵

(i) x³ – 6x² + x + 1 → Degree = 3
(ii) y⁹ – 3y⁷ + 3/2y² + 4 → Degree = 9
(iii) 3 – y³z → Sum of powers = 3 + 1 = 4. Degree = 4
(iv) x²y – 2x + 1 → Sum of powers = 2 + 1 = 3. Degree = 3
(v) 5t – 11 → Degree = 1
(vi) 7 → Constant polynomial 7 = 7x⁰. Degree = 0

Expressions: (i) x³ + x² + x + 1, (ii) 9x³, (iii) y + y² + 3/4, (iv) t + 3, (v) y – y³, (vi) 8, (vii) 2x² + 3, (viii) P² – P + 5, (ix) x + 2/3, (x) 4, (xi) -u/4 + 3/2, (xii) -3/7.
Categorization / :
Constant Polynomials: (vi) 8, (x) 4, (xii) -3/7
Linear Polynomials: (iv) t + 3, (ix) x + 2/3, (xi) -u/4 + 3/2
Binomials: (v) y – y³, (vii) 2x² + 3
Trinomials: (iii) y + y² + 3/4, (viii) P² – P + 5

(i) x = 0
Solution / : p(0) = 5(0)³ – 2(0)² + 3(0) – 2 = -2
(ii) x = 1
Solution / : p(1) = 5(1)³ – 2(1)² + 3(1) – 2 = 5 – 2 + 3 – 2 = 4
(iii) x = -2
Solution / : p(-2) = 5(-2)³ – 2(-2)² + 3(-2) – 2 = 5(-8) – 2(4) – 6 – 2 = -40 – 8 – 6 – 2 = -56

(i) p(x) = 4x³ + 2x² – 3x + 2
p(0) = 4(0) + 2(0) – 3(0) + 2 = 2
p(-1) = 4(-1)³ + 2(-1)² – 3(-1) + 2 = -4 + 2 + 3 + 2 = 3
p(2) = 4(2)³ + 2(2)² – 3(2) + 2 = 32 + 8 – 6 + 2 = 36
p(3) = 4(3)³ + 2(3)² – 3(3) + 2 = 108 + 18 – 9 + 2 = 119
(ii) p(r) = (r – 1)(r + 1)
p(0) = (0 – 1)(0 + 1) = -1
p(-1) = (-1 – 1)(-1 + 1) = 0
p(2) = (2 – 1)(2 + 1) = 1 × 3 = 3
p(3) = (3 – 1)(3 + 1) = 2 × 4 = 8
(iii) p(t) = 2/3t² – 1/3t + 1/3
p(0) = 0 – 0 + 1/3 = 1/3
p(-1) = 2/3(-1)² – 1/3(-1) + 1/3 = 2/3 + 1/3 + 1/3 = 4/3
p(2) = 2/3(4) – 1/3(2) + 1/3 = 8/3 – 2/3 + 1/3 = 7/3
p(3) = 2/3(9) – 1/3(3) + 1/3 = 6 – 1 + 1/3 = 5 + 1/3 = 16/3
(iv) p(y) = (y² – y + 1)(y + 1)
p(0) = (1)(1) = 1
p(-1) = (1 + 1 + 1)(-1 + 1) = 3 × 0 = 0
p(2) = (4 – 2 + 1)(2 + 1) = 3 × 3 = 9
p(3) = (9 – 3 + 1)(3 + 1) = 7 × 4 = 28
(v) p(x) = x + 2
p(0) = 0 + 2 = 2
p(-1) = -1 + 2 = 1
p(2) = 2 + 2 = 4
p(3) = 3 + 2 = 5

(i) p(x) = 3x + 1; x = -1/3
p(-1/3) = 3(-1/3) + 1 = -1 + 1 = 0 ⇒ *Yes
(ii) p(x) = x + 2; x = -2
p(-2) = -2 + 2 = 0 ⇒ *Yes
(iii) p(x) = 5x – 4; x = 5/4
p(5/4) = 5(5/4) – 4 = 25/4 – 4 = 9/4 ≠ 0 ⇒ *No
(iv) p(y) = y² – 1; y = 1, -1
p(1) = 1² – 1 = 0; p(-1) = (-1)² – 1 = 0 ⇒ *Yes
(v) p(t) = (t + 1)(t – 2); t = +1, -2
p(1) = (2)(-1) = -2 ≠ 0 ⇒ *No
(vi) p(x) = lx + m; x = -m/l
p(-m/l) = l(-m/l) + m = -m + m = 0 ⇒ *Yes
(vii) p(r) = 3r² – 1; r = -1/√3, 1/√3
p( ± 1/√3) = 3(1/3) – 1 = 1 – 1 = 0 ⇒ *Yes

(i) p(x) = x + 6 ⇒ x + 6 = 0 ⇒ x = -6
(ii) p(x) = x – 6 ⇒ x – 6 = 0 ⇒ x = 6
(iii) p(y) = 5y ⇒ 5y = 0 ⇒ y = 0
(iv) p(t) = at, a ≠ 0 ⇒ at = 0 ⇒ t = 0
(v) p(r) = cr + d, c ≠ 0 ⇒ cr + d = 0 ⇒ r = -d/c
(vi) p(u) = 3u – 6 ⇒ 3u = 6 ⇒ u = 2
(vii) r(s) = 2s + 3 ⇒ 2s = -3 ⇒ s = -3/2
(viii) p(x) = 5 – x ⇒ 5 – x = 0 ⇒ x = 5
(ix) q(t) = 1/2t – 2/3 ⇒ 1/2t = 2/3 ⇒ t = 4/3

(i) 2x² + x + 1 and 3x² + 4x + 5
(2x² + 3x²) + (x + 4x) + (1 + 5) = 5x² + 5x + 6
(ii) 8p² – 3p + 4 and 3p³ – 4p + 7
3p³ + 8p² + (-3p – 4p) + (4 + 7) = 3p³ + 8p² – 7p + 11
(iii) -5x³ + 9x² – 5x + 7 and -2x² + 7x³ – 3x – 8
(-5x³ + 7x³) + (9x² – 2x²) + (-5x – 3x) + (7 – 8) = 2x³ + 7x² – 8x – 1

(i) 3y² + 2y – 5; 2y² + 5 + 8y and -y² – y
Sum = (3 + 2 – 1)y² + (2 + 8 – 1)y + (-5 + 5) = 4y² + 9y
Degree: 2
(ii) 5 + 7r – 3r²; r² + 7 and r² – 3r + 5
Sum = (-3 + 1 + 1)r² + (7 – 3)r + (5 + 7 + 5) = -r² + 4r + 17
Degree: 2
(iii) 4x + 7 – 3x² + 5x³; 7x² – 2x + 1 and -2x³ – 2x
Sum = (5 – 2)x³ + (-3 + 7)x² + (4 – 2 – 2)x + (7 + 1) = 3x³ + 4x² + 8
Degree: 3

(i) t² – 5t + 2 from 7t³ – 3t² + 2
(7t³ – 3t² + 2) – (t² – 5t + 2) = 7t³ + (-3 – 1)t² + 5t + (2 – 2) = 7t³ – 4t² + 5t
(ii) 3p – 5p² + 7 + 3p³ from 2p² – 5 + 11p – p³
(-p³ + 2p² + 11p – 5) – (3p³ – 5p² + 3p + 7) = (-1 – 3)p³ + (2 + 5)p² + (11 – 3)p + (-5 – 7)
= -4p³ + 7p² + 8p – 12
(iii) 5z³ + 7z² + 2z – 4 from -3z² + 11z + 12z³ + 13
(12z³ – 3z² + 11z + 13) – (5z³ + 7z² + 2z – 4) = (12 – 5)z³ + (-3 – 7)z² + (11 – 2)z + (13 + 4)
= 7z³ – 10z² + 9z + 17

(i) (3x + 4) by (7x² + 2x + 1)
3x(7x² + 2x + 1) + 4(7x² + 2x + 1) = 21x³ + 6x² + 3x + 28x² + 8x + 4
= 21x³ + 34x² + 11x + 4
(ii) (5x³ + 2x) by (3x² – 9x + 6)
5x³(3x² – 9x + 6) + 2x(3x² – 9x + 6) = 15x⁵ – 45x⁴ + 30x³ + 6x³ – 18x² + 12x
= 15x⁵ – 45x⁴ + 36x³ – 18x² + 12x
(iii) (p⁴ – 5p² + 3) by (p³ + 1)
p³(p⁴ – 5p² + 3) + 1(p⁴ – 5p² + 3) = p⁷ – 5p⁵ + 3p³ + p⁴ – 5p² + 3
= p⁷ – 5p⁵ + p⁴ + 3p³ – 5p² + 3

Example 1 / 1: Find zero of p(x) = 2x + 1.
Solution / : Put 2x + 1 = 0 ⇒ 2x = -1 ⇒ x = -1/2.
Example 2 / 2: Check if 0 or 2 are zeroes of x² – 2x.
Solution / : p(0) = 0² – 2(0) = 0; p(2) = 2² – 2(2) = 0. Both 0 and 2 are zeroes.
Example 3 / 3: Add 3x³ – x² + 5x – 4 and 3x² – 7x + 8.
Solution / : 3x³ + (-1 + 3)x² + (5 – 7)x + (-4 + 8) = 3x³ + 2x² – 2x + 4.
Example 4 / 4: Add 3/2y³ + y² + y + 1 and y⁴ – 1/2y³ – 3y + 1.
Solution / : y⁴ + (3/2 – 1/2)y³ + y² + (1 – 3)y + (1 + 1) = y⁴ + y³ + y² – 2y + 2.
Example 5 / 5: Subtract 4x² + 3x – 2 from 9x² – 3x – 7.
Solution / : (9x² – 3x – 7) – (4x² + 3x – 2) = 5x² – 6x – 5.
Example 6 / 6: Subtract s(z) = 3z – 5z² + 7 + 3z³ from x(z) = -z³ + 2z² + 11z – 5.
Solution / : (-z³ + 2z² + 11z – 5) – (3z³ – 5z² + 3z + 7) = -4z³ + 7z² + 8z – 12.
Example 7 / 7: Add 3x + 4 – 5x², 5 + 9x, and 4x – 17 – 5x². Find degree.
Solution / : Sum = -10x² + 16x – 8. Degree = 2.
Example 8 / 8: Subtract r(x) = 2x² – 3x – 1 from sum of p(x) = 3x² – 8x + 11 and q(x) = -4x² + 15. Find degree.
Solution / : p(x) + q(x) = -x² – 8x + 26. Difference = (-x² – 8x + 26) – (2x² – 3x – 1) = -3x² – 5x + 27. Degree = 2.
Example 9 / 9: Find sum and difference of p(x) = 4x³ + 3x² + 2x – 1 and q(x) = 4x³ + 2x² – 2x + 5.
Solution / :
Sum: 8x³ + 5x² + 4
Difference: x² + 4x – 6
Example 10 / 10: What should be added to 2u² – 4u + 3 to get 4u³ – 5u² + 1?
Solution / : (4u³ – 5u² + 1) – (2u² – 4u + 3) = 4u³ – 7u² + 4u – 2.
Example 11 / 11: What should be subtracted from 2y³ – 3y² + 4 to get y³ – 1?
Solution / : (2y³ – 3y² + 4) – (y³ – 1) = y³ – 3y² + 5.
Example 12 / 12: Multiply 2x² + 3x + 4 by 3.
Solution / : 3(2x² + 3x + 4) = 6x² + 9x + 12.
Example 13 / 13: Multiply (2x + 5) × (4x + 3).
Solution / : 2x(4x + 3) + 5(4x + 3) = 8x² + 6x + 20x + 15 = 8x² + 26x + 15.
Example 14 / 14: Multiply (2x + 5) × (3x² + 4x + 6), find degree.
Solution / : Product = 6x³ + 23x² + 32x + 30. Degree = 3.
Example 15 / 15: Find p(x) · q(x) for p(x) = 2x + 3 and q(x) = x² + x – 2.
Solution / : (2x + 3)(x² + x – 2) = 2x³ + 5x² – x – 6.

💡 Solution:

हल / हल: It is a polynomial because the exponent of variable x in all terms is a whole number (2, 1, 0). / यह एक बहुपद है क्योंकि चर x की घातें पूर्ण संख्याएँ हैं।
हल / हल: It is not a polynomial because z + 3/z = z + 3z^-1, where the power of z is -1, which is not a whole number. / यह बहुपद नहीं है क्योंकि z + 3z^-1 में चर z की घात -1 है जो पूर्ण संख्या नहीं है।
हल / हल: It is not a polynomial because √y = y^1/2, where the exponent is 1/2, which is not a whole number. / यह बहुपद नहीं है क्योंकि चर y की घात 1/2 है जो पूर्ण संख्या नहीं है।
हल / हल: It is a polynomial because the powers of variable x are whole numbers. / यह एक बहुपद है क्योंकि चर x की घातें पूर्ण संख्याएँ हैं।
Solution / हल: It is a polynomial because the powers of variables x, y, t are whole numbers (10, 3, 50). / यह एक बहुपद है क्योंकि चरों की घातें पूर्ण संख्याएँ हैं।

हल / हल: Combining 3x² + 2x² = 5x², the का गुणांक x² is 5 (or 2 if 3x³ term). / x² का गुणांक 5 है।
हल / हल: का गुणांक x² is 3. / x² का गुणांक 3 है।
हल / हल: का गुणांक x² is -5. / x² का गुणांक -5 है।
हल / हल: का गुणांक x² is 1/2. / x² का गुणांक 1/2 है।
हल / हल: का गुणांक x² is 1/4. / x² का गुणांक 1/4 है।

हल / हल: का गुणांक x = 1/5, अचर पद = 5. / x का गुणांक = 1/5, अचर पद = 5।
हल / हल: का गुणांक x = 0, अचर पद = 7. / x का गुणांक = 0, अचर पद = 7।
हल / हल: का गुणांक x = 2, अचर पद = 0. / x का गुणांक = 2, अचर पद = 0।

Solution / :
Sum = (x⁴ + 3x³ + 2x + 6) + (x⁴ – 3x² + 6x + 2) = 2x⁴ + 3x³ – 3x² + 8x + 8
Subtracting x³ – 3x + 4:
(2x⁴ + 3x³ – 3x² + 8x + 8) – (x³ – 3x + 4) = 2x⁴ + 2x³ – 3x² + 11x + 4

Solution / :
p(u) + q(u) = (u⁷ – u⁷) – u⁵ + 2u² + u + (1 – 2) = -u⁵ + 2u² + u – 1
Degree: 5

Solution / :
Required polynomial = (x² – x + 4) – (x³ – 3x² + 6) = -x³ + (1 + 3)x² – x + (4 – 6)
= -x³ + 4x² – x – 2

Solution / :
Required polynomial = (u⁶ – u – 4) – (u⁷ – 3u⁶ + 4u² + 2) = -u⁷ + (1 + 3)u⁶ – 4u² – u + (-4 – 2)
= -u⁷ + 4u⁶ – 4u² – u – 6

Solution / :
Required polynomial = (y³ – 3y² + y + 2) – (y³ + 2y + 1) = (1 – 1)y³ – 3y² + (1 – 2)y + (2 – 1)
= -3y² – y + 1

Solution / :
Required polynomial = (t² + t – 7) – (t³ + t² + 3t + 4) = -t³ + (1 – 1)t² + (1 – 3)t + (-7 – 4)
= -t³ – 2t – 11

Solution / :
p(x) · q(x) = (x³ + 7x + 3)(2x³ – 3) = x³(2x³ – 3) + 7x(2x³ – 3) + 3(2x³ – 3)
= 2x⁶ – 3x³ + 14x⁴ – 21x + 6x³ – 9
= 2x⁶ + 14x⁴ + 3x³ – 21x – 9

Solution / :
Degree of p(u) = 2
Degree of q(u) = 2
Degree of r(u) = 1
Degree of product p(u)q(u)r(u) = 2 + 2 + 1 = 5

CG Board Class 9 Mathematics Chapter 4 Solutions: Polynomials (बहुपद) – 2026-27

End-of-Chapter Exercise 4.1 (प्रश्नावली 4.1)

End-of-Chapter Exercises (Solutions)

अभ्यास के सभी प्रश्नों के हल

(i) 4×2 – 3x + 5 Solution / : It is a polynomial because the…

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End-of-Chapter Exercise 4.1 (प्रश्नावली 4.1)

End-of-Chapter Exercises (Solutions)

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(i) 4×2 – 3x + 5 Solution / : It is a polynomial because the…

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