CBSE Important Questions Class 10 Maths Chapter 2 – Polynomials

Class 10 Maths Chapter 2 Polynomials is part of the CBSE Class 10 Maths syllabus. It explains what a polynomial is, what the degree of a polynomial means, and what the zeroes of a polynomial are. Students learn how to read the zeroes from the graph of a polynomial, how the zeroes of a quadratic polynomial are related to its coefficients, and how to build a quadratic polynomial when the sum and product of its zeroes are known.

These CBSE important questions for Class 10 Maths Chapter 2 cover finding the number of zeroes from a graph, the relation between the zeroes and the coefficients, finding the value of α² + β², forming a polynomial from the sum and product of its zeroes, and solving for an unknown constant when a condition on the zeroes is given. Every question has a clear step-by-step solution in simple English with figures, so students can practise on their own, parents can check the working, and teachers can use them for class tests.

Q1. Observe the graph of the polynomial x² + x − 3 shown below. Determine the number of zeroes of the polynomial and explain your reasoning based on the graph.

Answer: The polynomial has two zeroes.
The zeroes of a polynomial are the points where its graph cuts the x-axis, because at those points the value of the polynomial is 0. In the given graph the curve crosses the x-axis at two distinct points, so the polynomial has two zeroes.

Q2. According to the graph, the number of zeroes for the polynomial x² + x − 3 will be
(a) zero
(b) one
(c) two
(d) three

Answer: (c) two
The graph of a quadratic polynomial is a parabola. Here it cuts the x-axis at two separate points, so there are two zeroes. A quadratic polynomial can never have more than two zeroes, because its degree is 2.

Q3. If α and β are the zeroes of the quadratic polynomial ax² + bx + c, write the value of α + β and αβ.

Answer: For a quadratic polynomial ax² + bx + c (where a ≠ 0):
Sum of the zeroes: α + β = −b/a  (that is, − coefficient of x ÷ coefficient of x²)
Product of the zeroes: αβ = c/a  (that is, constant term ÷ coefficient of x²)
These two relations are used in almost every question of this chapter.

Q4. If α and β are the roots of the equation 2ax² + bx + c = 0, where a ≠ 0, find the value of (α² + β²).

Answer: For the equation 2ax² + bx + c = 0:
α + β = −b / 2a  and  αβ = c / 2a
We use the identity α² + β² = (α + β)² − 2αβ
= (−b/2a)² − 2(c/2a)
= b²/4a² − c/a
Taking the LCM 4a²:
= (b² − 4ac) / 4a²
So α² + β² = (b² − 4ac) / 4a²

Q5. If α and β are the zeroes of the polynomial f(x) = x² − p(x + 1) − c such that (α + 1)(β + 1) = 0, then the value of c is
(a) 0
(b) 1
(c) 2
(d) 3

Answer: (b) 1
First write the polynomial in standard form:
f(x) = x² − px − p − c
So α + β = p and αβ = −p − c.
Now expand the given condition:
(α + 1)(β + 1) = αβ + α + β + 1 = 0
Substituting: (−p − c) + p + 1 = 0
−c + 1 = 0
So c = 1.

Q6. Find the zeroes of the quadratic polynomial f(x) = x² − 3x − 28 and verify the relationship between the zeroes and the coefficients.

Answer: Split the middle term. We need two numbers whose product is −28 and whose sum is −3. These are −7 and +4.
x² − 7x + 4x − 28 = 0
x(x − 7) + 4(x − 7) = 0
(x − 7)(x + 4) = 0
So the zeroes are α = 7 and β = −4.
Verification: Here a = 1, b = −3, c = −28.
Sum of zeroes = 7 + (−4) = 3, and −b/a = −(−3)/1 = 3 ✓
Product of zeroes = 7 × (−4) = −28, and c/a = −28/1 = −28 ✓

Q7. Find the zeroes of the quadratic polynomial x² − 2x − 8 and verify the relationship between the zeroes and the coefficients.

Answer: Split the middle term. We need two numbers whose product is −8 and whose sum is −2. These are −4 and +2.
x² − 4x + 2x − 8 = 0
x(x − 4) + 2(x − 4) = 0
(x − 4)(x + 2) = 0
So the zeroes are α = 4 and β = −2.
Verification: Here a = 1, b = −2, c = −8.
Sum of zeroes = 4 + (−2) = 2, and −b/a = −(−2)/1 = 2 ✓
Product of zeroes = 4 × (−2) = −8, and c/a = −8/1 = −8 ✓

Q8. Find the zeroes of the quadratic polynomial f(x) = abx² + (b² − ac)x − bc, where a ≠ 0 and b ≠ 0, and verify the relationship between the zeroes and the coefficients.

Answer: Split the middle term:
abx² + b²x − acx − bc = 0
bx(ax + b) − c(ax + b) = 0
(ax + b)(bx − c) = 0
So the zeroes are x = −b/a and x = c/b.
Verification: Coefficients are A = ab, B = b² − ac, C = −bc.
Sum of zeroes = −b/a + c/b = (−b² + ac)/ab = −(b² − ac)/ab = −B/A ✓
Product of zeroes = (−b/a) × (c/b) = −c/a = −bc/ab = C/A ✓

Q9. Find a quadratic polynomial whose sum of zeroes is 4 and product of zeroes is 1.

Answer: If the sum and product of the zeroes are known, the quadratic polynomial is:
p(x) = x² − (sum of zeroes)x + (product of zeroes)
Substituting the given values:
p(x) = x² − 4x + 1
So the required polynomial is x² − 4x + 1 (or any constant multiple of it).

Q10. If one zero of the quadratic polynomial x² − 5x + k is 2, find the value of k and the other zero.

Answer: Since 2 is a zero, putting x = 2 must make the polynomial equal to zero.
(2)² − 5(2) + k = 0
4 − 10 + k = 0
k = 6
Other zero: Sum of the zeroes = −b/a = 5.
So 2 + other zero = 5, which gives the other zero = 3.
(Check: product of zeroes 2 × 3 = 6 = c/a ✓)

Q11. If α and β are the zeroes of the polynomial x² − 6x + 8, find the value of (1/α + 1/β).

Answer: For x² − 6x + 8: a = 1, b = −6, c = 8.
α + β = −b/a = 6
αβ = c/a = 8
Now, 1/α + 1/β = (β + α) / αβ
= 6 / 8
= 3/4

Q12. What is the degree of a polynomial? How many zeroes can a linear, quadratic and cubic polynomial have?

Answer: The degree of a polynomial is the highest power of the variable in it. For example, the degree of 3x² + 5x − 2 is 2.
Number of zeroes: A polynomial of degree n can have at most n zeroes.
• A linear polynomial (degree 1), such as 2x + 3, has exactly one zero.
• A quadratic polynomial (degree 2) has at most two zeroes.
• A cubic polynomial (degree 3) has at most three zeroes.
On a graph, the number of zeroes is the number of points where the curve cuts the x-axis.

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