Welcome to AssamSchool. This article contains 30 Extra Questions from Polynomials from Class 9 Mathematics Chapter 2. These Polynomials Class 9 Extra Questions are helpful to test your understanding and knowledge of the chapter. These additional questions will not only help you to excel in the chapter but also will give you a strong foundation in Mathematics.
We have also provided step-by-step answers to the questions for reference. So, let’s start…
30 Polynomials Class 9 Extra Questions
1. Determine whether the expression \(3x^2 + \sqrt{5}x – 7\) is a polynomial. Justify your answer.
Solution:
Yes, it is a polynomial. All exponents of \(x\) are whole numbers \(2, 1,\) and \(0\), and the coefficients \(3\), \(\sqrt{5}\), and \(-7\) are constants.
2. Find the coefficient of \(x^3\) in the polynomial \(p(x) = 2 – x^3 + 4x^5 – 6x\)
Solution:
The term with \(x^3\) is \(-x^3\)
Thus, the coefficient is \(-1\)
3. Verify if \(x = 2\) is a zero of the polynomial \(p(x) = x^3 – 3x^2 + 4x – 12\)
Solution:
Substitute \(x = 2\):
\(p(2) = 2^3 – 3(2)^2 + 4(2) – 12 = 8 – 12 + 8 – 12 = -8 \neq 0\)
Hence, \(x = 2\) is not a zero.
4. Check if (\(x + 1)\) is a factor of \(p(x) = x^4 + x^3 – x – 1\)
Solution:
Find \(p(-1)\):
\(p(-1) = (-1)^4 + (-1)^3 – (-1) – 1 = 1 – 1 + 1 – 1 = 0\)
Since \(p(-1) = 0\), (\(x + 1)\) is a factor.
5. Expand ((2x – 3y)^2) using identities.
Solution:
Using (\(a – b)^2 = a^2 – 2ab + b^2\):
\((2x – 3y)^2 = (2x)^2 – 2(2x)(3y) + (3y)^2 = 4x^2 – 12xy + 9y^2\)
6. Factorise \(6x^2 + 17x + 5\) by splitting the middle term.
Solution:
Find two numbers \(a\) and \(b\) such that \(a + b = 17\) and \(ab = 6 \times 5 = 30\)
Numbers: 15 and 2.
\(6x^2 + 15x + 2x + 5 = 3x(2x + 5) + 1(2x + 5) = (3x + 1)(2x + 5)\)
7. The area of a rectangle is \(12x^2 – 7x – 10\). Find its possible dimensions.
Solution:
Factorise \(12x^2 – 7x – 10\):
Find \(a\) and \(b\) such that \(a + b = -7\), \(ab = 12 \times (-10) = -120\)
Numbers: \(-15\) and \(8\)
\(12x^2 – 15x + 8x – 10 = 3x(4x – 5) + 2(4x – 5) = (3x + 2)(4x – 5)\)
Dimensions: \((3x + 2)\) and (\(4x – 5)\)
8. Is \(y + \frac{1}{y}\) a polynomial? Why?
Solution:
No. The term \(\frac{1}{y} = y^{-1}\) has a negative exponent, which is not a whole number.
9. Write the degree of the polynomial \(p(t) = 5t^7 – t^4 + 9\)
Solution:
The highest power of \(t\) is \(7\). Hence, degree = \(7\)
10. Find the zero of the polynomial \(p(x) = 7x – 21\)
Solution:
Set \(7x – 21 = 0 \implies x = 3\)
Zero: \(x = 3\)
11. If (\(x – 2)\) is a factor of \(x^3 – kx^2 + 4x – 8\), find \(k\).
Solution:
Substitute \(x = 2\):
\(2^3 – k(2)^2 + 4(2) – 8 = 8 – 4k + 8 – 8 = 8 – 4k = 0 \implies k = 2\)
12. Evaluate \(103^2\) using an identity.
Solution:
\(103^2 = (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609\)
13. Factorise \(x^3 – 3x^2 – 9x – 5\) using the factor theorem.
Solution:
Test \(x = -1\):
\((-1)^3 – 3(-1)^2 – 9(-1) – 5 = -1 – 3 + 9 – 5 = 0\)
Thus, (\(x + 1)\) is a factor. Divide and factor further:
\(x^3 – 3x^2 – 9x – 5 = (x + 1)(x^2 – 4x – 5) = (x + 1)(x – 5)(x + 1)\)
14. The volume of a cuboid is \(2x^3 + 16x^2 + 30x\). Find its dimensions.
Solution:
Factorise:
\(2x(x^2 + 8x + 15) = 2x(x + 3)(x + 5)\)
Dimensions: \(2x\), (\(x + 3)), ((x + 5)\)
15. Is \(\sqrt{3}x^2 + 5x – 1\) a polynomial? Justify.
Solution:
Yes. All exponents are whole numbers, and \(\sqrt{3}\) is a constant coefficient.
16. Identify the constant term in \(p(y) = y^4 – 2y^3 + 7\)
Solution:
The constant term is \(7\).
17. If \(x = 1\) is a zero of \(p(x) = ax^3 – 4x + 3\), find \(a\).
Solution:
Substitute \(x = 1\):
\(a(1)^3 – 4(1) + 3 = a – 4 + 3 = a – 1 = 0 \implies a = 1\)
18. Show that (\(x – 3)\) is a factor of \(x^3 – 6x^2 + 11x – 6\).
Solution:
Calculate \(p(3)\):
\(3^3 – 6(3)^2 + 11(3) – 6 = 27 – 54 + 33 – 6 = 0\)
Hence, (\(x – 3)\) is a factor.
19. Expand (\(x + 2y + 3z)^2\).
Solution:
Using (\(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\):
\(x^2 + (2y)^2 + (3z)^2 + 2(x)(2y) + 2(2y)(3z) + 2(x)(3z) \)
\(= x^2 + 4y^2 + 9z^2 + 4xy + 12yz + 6xz\)
20. Factorise \(8m^3 – 125n^3\).
Solution:
Using \(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\):
\((2m)^3 – (5n)^3 = (2m – 5n)(4m^2 + 10mn + 25n^2)\)
21. The area of a square is \(9x^2 + 24x + 16\). Find its side length.
Solution:
Factorise:
\(9x^2 + 24x + 16 = (3x + 4)^2\)
Side length = \(3x + 4\).
22. Why is \(x^{3/2} + 2x\) not a polynomial?
Solution:
The term \(x^{3/2}\) has a fractional exponent (\(\frac{3}{2})\), which is not a whole number.
23. Write a monomial of degree 50.
Solution:
A monomial of degree 50 is \(7x^{50}\)
24. Find all zeroes of \(p(x) = x^2 – 5x + 6\).
Solution:
Factorise:
\(x^2 – 5x + 6 = (x – 2)(x – 3) \implies \text{Zeroes: } x = 2, 3\)
25. Find \(k\) if (\(x + 2)\) is a factor of \(x^3 + 4x^2 + kx + 8\).
Solution:
Substitute \(x = -2\):
\((-2)^3 + 4(-2)^2 + k(-2) + 8 = -8 + 16 – 2k + 8 = 16 – 2k = 0 \implies k = 8\)
26. Evaluate \(98 \times 102\) using an identity.
Solution:
\(98 \times 102 = (100 – 2)(100 + 2) = 100^2 – 2^2 = 10000 – 4 = 9996\)
27. Factorise \(x^3 + 8y^3 + 6x^2y + 12xy^2\).
Solution:
Using (\(a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2\):
\(x^3 + (2y)^3 + 3x^2(2y) + 3x(2y)^2 = (x + 2y)^3\)
28. The volume of a cuboid is (x^3 – 6x^2 + 11x – 6). Find its dimensions.
Solution:
Factorise:
\(x^3 – 6x^2 + 11x – 6 = (x – 1)(x – 2)(x – 3)\)
Dimensions: (\(x – 1)\), (\(x – 2)\), (\(x – 3)\).
29. Is \(\frac{1}{x} + x^2\) a polynomial? Why?
Solution:
No. The term \(\frac{1}{x} = x^{-1}\) has a negative exponent.
30. Expand (\(a – b – c)^2\).
Solution:
Using (\(x + y + z)^2\) with \(x = a\), \(y = -b\), \(z = -c\):
\(a^2 + (-b)^2 + (-c)^2 + 2(a)(-b) + 2(-b)(-c) + 2(a)(-c)\)
\( = a^2 + b^2 + c^2 – 2ab + 2bc – 2ac\)
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Abdur Rohman is an Electrical Engineer from Charaideo, Assam, who wears multiple hats as a part-time teacher, blogger, entrepreneur, and digital marketer. Passionate about education, he founded The Assam School blog to provide free, comprehensive textbook solutions, MCQs (Multiple Choice Questions), and other academic content for students from Class V to XII.