Class 10 · Maths

Polynomials

This chapter studies the zeroes of a polynomial and how they are related to its coefficients. It covers the geometrical meaning of zeroes, the relationship between zeroes and coefficients for quadratic and cubic polynomials, and the division algorithm.

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Exercise 2.1

Q1. Exercise 2.1 — Question 1

The graph of a quadratic polynomial cuts the x-axis at two distinct points. How many zeroes does it have, and why?

The zeroes of a polynomial are exactly the x-coordinates of the points where its graph meets the x-axis, because at those points the value of the polynomial is 0. Here the graph crosses the x-axis at two distinct points, so there are two distinct values of x for which p(x) = 0. This also agrees with the fact that a quadratic polynomial can have at most 2 zeroes.

Final Answer: It has 2 zeroes — one for each point where the graph meets the x-axis.

Common Mistake: Counting the point where the graph cuts the y-axis as a zero.

Exam Tip: Zeroes ↔ x-axis intersections. A quadratic has at most 2, a cubic at most 3.

Exercise 2.2

Q1. Exercise 2.2 — Question 1

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

Factorise by splitting the middle term. We need two numbers whose product is −8 and whose sum is −2: these are −4 and +2. x² − 2x − 8 = x² − 4x + 2x − 8 = x(x − 4) + 2(x − 4) = (x − 4)(x + 2) Setting each factor to zero gives x = 4 and x = −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 ✓

Formula: For ax² + bx + c: sum of zeroes = −b/a, product = c/a

Final Answer: Zeroes are 4 and −2; sum = 2 = −b/a and product = −8 = c/a.

Common Mistake: Forgetting the minus sign in −b/a when computing the sum.

Exam Tip: The verification step carries marks — always show both the sum and the product check.

Exercise 2.3

Q1. Exercise 2.3 — Question 1

Apply the division algorithm to divide 2x³ + 3x² − x + 1 by (x + 2), and state the quotient and remainder.

Divide step by step: 2x³ ÷ x = 2x². Multiply: 2x²(x + 2) = 2x³ + 4x². Subtract: (3x² − 4x²) = −x². Bring down the next term: −x² − x. −x² ÷ x = −x. Multiply: −x(x + 2) = −x² − 2x. Subtract: (−x + 2x) = x. Bring down +1: x + 1. x ÷ x = 1. Multiply: 1(x + 2) = x + 2. Subtract: (1 − 2) = −1. So the quotient is 2x² − x + 1 and the remainder is −1. Check with the division algorithm: (x + 2)(2x² − x + 1) + (−1) = 2x³ + 3x² − x + 2 − 1 = 2x³ + 3x² − x + 1 ✓

Formula: Division algorithm: p(x) = g(x) × q(x) + r(x)

Final Answer: Quotient = 2x² − x + 1, Remainder = −1

Common Mistake: Sign slips while subtracting each partial product — subtract the whole bracket, not just the first term.

Exam Tip: Always verify with p(x) = g(x)q(x) + r(x); it catches mistakes and earns a mark.

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