CBSE Class 10 Maths — Polynomials

Chapter Overview & Weightage

Polynomials is a foundational chapter in CBSE Class 10 Maths that bridges algebra and geometry. The graphical interpretation of zeroes and the relationship between zeroes and coefficients are the two pillars of this chapter.

In CBSE Class 10 board exams, Polynomials typically carries 5–8 marks out of the 80-mark paper. Questions include finding zeroes from graphs (1 mark), writing polynomials with given zeroes (2 marks), and verifying/using the sum-product relationship (2–3 marks). This chapter is highly formula-driven and completely scorable.

Topics covered:

Geometrical meaning of zeroes of a polynomial
Relationship between zeroes and coefficients (quadratic and cubic)
Division algorithm for polynomials

Key Concepts You Must Know

1. Zeroes of a Polynomial A zero (root) of polynomial $p(x)$ is a value $\alpha$ such that $p(\alpha) = 0$ . Geometrically, zeroes are the x-coordinates where the graph of $y = p(x)$ crosses (or touches) the x-axis.

2. Number of Zeroes

A linear polynomial has exactly 1 zero
A quadratic polynomial has at most 2 zeroes
A cubic polynomial has at most 3 zeroes
An $n$ -degree polynomial has at most $n$ zeroes

3. Graphical meaning:

Cuts x-axis: zero with odd multiplicity
Touches x-axis (doesn’t cross): zero with even multiplicity

4. Relationship between zeroes and coefficients (quadratic): For $p(x) = ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$ :

\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}

5. Relationship for cubic: For $p(x) = ax^3 + bx^2 + cx + d$ with zeroes $\alpha, \beta, \gamma$ :

\alpha + \beta + \gamma = -\frac{b}{a}

\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}

\alpha\beta\gamma = -\frac{d}{a}

Important Formulas

For $ax^2 + bx + c = 0$ with zeroes $\alpha, \beta$ :

Sum: $\alpha + \beta = -b/a$

Product: $\alpha\beta = c/a$

Forming polynomial: $p(x) = x^2 - (\alpha+\beta)x + \alpha\beta$

For polynomials $p(x)$ and $g(x)$ (where $g(x) \neq 0$ ):

p(x) = g(x) \cdot q(x) + r(x)

where $r(x) = 0$ or $\deg(r) < \deg(g)$

Solved Previous Year Questions

PYQ 1 — Find zeroes and verify (CBSE 2023 Style)

Find the zeroes of $p(x) = x^2 - 5x + 6$ and verify the relationship between zeroes and coefficients.

Solution: $x^2 - 5x + 6 = (x-2)(x-3) = 0$

Zeroes: $\alpha = 2$ , $\beta = 3$

Verification: Sum of zeroes = $2 + 3 = 5 = -(-5)/1 = -b/a$ ✓

Product of zeroes = $2 \times 3 = 6 = 6/1 = c/a$ ✓

PYQ 2 — Form a quadratic polynomial

Form a quadratic polynomial whose zeroes are $-3$ and $4$ .

Solution: Sum of zeroes = $-3 + 4 = 1$

Product of zeroes = $(-3)(4) = -12$

$p(x) = x^2 - (\text{sum})x + (\text{product}) = x^2 - x - 12$

Check: $(-3)^2 - (-3) - 12 = 9 + 3 - 12 = 0$ ✓

PYQ 3 — Cubic polynomial (CBSE 2024 Style)

If two zeroes of $x^3 - 4x^2 - 3x + 12$ are $\sqrt{3}$ and $-\sqrt{3}$ , find the third zero.

Solution: Sum of all three zeroes = $-b/a = -(-4)/1 = 4$

Let the third zero be $\gamma$ .

$\sqrt{3} + (-\sqrt{3}) + \gamma = 4$

$0 + \gamma = 4$

$\gamma = 4$

Verification by product: $\sqrt{3} \times (-\sqrt{3}) \times 4 = -3 \times 4 = -12 = -d/a = -12/1$ ✓

Difficulty Distribution

Question Type	Marks	Difficulty	Frequency
Find zeroes of quadratic	1–2	Easy	High
Verify sum-product relationship	2	Easy	High
Form polynomial from zeroes	2	Easy	High
Third zero of cubic	3	Medium	Medium
Division algorithm	3–4	Medium	Medium
Graph-based zero identification	1	Easy	Medium

About 80% of Polynomials marks are from straightforward formula-application — this is one of the highest-scoring chapters if you practise enough.

Expert Strategy

Master the two formulas cold. Sum = $-b/a$ , Product = $c/a$ for quadratic. Write them 20 times if needed. In the exam, apply them without thinking.

Always verify your zeroes by substituting back into the polynomial. If your zero doesn’t give $p(\alpha) = 0$ , you’ve made an error — find it before moving on.

For cubic polynomials, the CBSE approach is always: two zeroes are given, find the third using the sum formula. You don’t need to solve a cubic from scratch.

When forming a quadratic with given zeroes $\alpha$ and $\beta$ , remember the answer is not unique — you can always multiply by any non-zero constant $k$ . The standard form is $k(x^2 - (\alpha+\beta)x + \alpha\beta)$ . Unless asked for a specific $k$ , use $k = 1$ .

Common Traps

Trap 1 — Sign error in sum of zeroes: The formula is $\alpha + \beta = -b/a$ , NOT $+b/a$ . If $p(x) = x^2 - 5x + 6$ , then $b = -5$ , and sum = $-(-5)/1 = +5$ . Students who memorise carelessly write sum = $b/a = -5$ , which is wrong.

Trap 2 — Division Algorithm confusion: $p(x) = g(x) \cdot q(x) + r(x)$ . Students sometimes write this as $p(x) \div g(x) = q(x) + r(x)$ , forgetting that remainder must be added after multiplying $g(x) \times q(x)$ . Always write it in the full form and verify by expanding.

Trap 3 — “Touches axis” vs “crosses axis”: If a quadratic graph touches the x-axis without crossing, both zeroes are equal (repeated root). For example, $p(x) = (x-3)^2$ has a double zero at $x = 3$ and touches the axis. Students sometimes count two separate zeroes here — there is only one distinct zero (with multiplicity 2).