Is it correct to say: Forgetting the minus sign in α + β = −b/a?

The minus sign is essential. Sum is −b/a, product is c/a.

Is it correct to say: Writing polynomial as ax² + bx + c when 'a' is missing?

Highest-degree coefficient MUST be non-zero. Otherwise the degree drops.

Is it correct to say: Saying degree of √x = 1?

A polynomial only allows non-negative integer exponents. √x = x^(1/2) is NOT a polynomial.

Is it correct to say: Confusing zero of polynomial with y-intercept?

Zero = x-value at which p(x) = 0 (x-axis crossing). y-intercept = p(0).

Is it correct to say: Assuming the number of zeroes always equals the degree over the reals?

Over reals a polynomial has AT MOST 'degree' zeroes; some may be complex or repeated.

Is it correct to say: Applying αβ = c/a to a cubic?

For a cubic use αβ + βγ + γα = c/a. αβγ = −d/a. Learn the three cubic relations.

Polynomials

Degree, zeroes, geometrical meaning, and relation between zeroes and coefficients for linear, quadratic and cubic polynomials — NCERT Class 10 Maths Ch 2

📐 10 formulas✏️ 3 examples🎯 6 practice⚖️ 3-4 marks🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→Sum and product of zeroes relation (α + β = −b/a, αβ = c/a) — asked almost every year, 2-3 marks.
→Form the quadratic given zeroes: x² − (α + β)x + αβ = 0.
→Graph-based question: read the number of zeroes from where the graph cuts the x-axis.
→For cubic, use αβ + βγ + γα = c/a and αβγ = −d/a — memorise the signs carefully.
→Division algorithm p(x) = g(x)·q(x) + r(x) is a HOTS favourite.

📐 Formulas(10)

Standard Form of a Polynomial

p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}

Zero of a Polynomial★ Board fav

p (α) = 0 ⟺ α is a zero of p (x)

Zeroes of a Quadratic — Sum★ Board fav

α + β = - \frac{b}{a}

Zeroes of a Quadratic — Product★ Board fav

α β = \frac{c}{a}

Form Quadratic from Zeroes

x^{2} - (α + β) x + α β = 0

Cubic — Sum of Zeroes

α + β + γ = - \frac{b}{a}

Cubic — Sum of Products (Two at a Time)

α β + β γ + γ α = \frac{c}{a}

Cubic — Product of Zeroes

α β γ = - \frac{d}{a}

Division Algorithm for Polynomials★ Board fav

p (x) = g (x) q (x) + r (x) with de g r (x) < de g g (x)

Factor Theorem

(x - α) divides p (x) ⟺ p (α) = 0

✏️ Solved Examples

1Solved Exampleeasy4 steps

Find the zeroes of the polynomial p(x) = x² − 5x + 6 and verify the relation between zeroes and coefficients.

Factorise

x^{2} - 5 x + 6 = (x - 2) (x - 3)

2Solved Exampleboard4 steps

If the sum of the zeroes of the quadratic 2x² + kx + 3 = 0 is 4, find k. Also find the product of zeroes.

Compare with ax² + bx + c: a = 2, b = k, c = 3.

3Solved ExampleHOTS3 steps

If two zeroes of the cubic x³ − 4x² + 5x − 2 are 1 and 1, find the third zero.

Let zeroes be α = 1, β = 1, γ = ?. Use sum = −b/a

α + β + γ = - \frac{- 4}{1} = 4

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Forgetting the minus sign in α + β = −b/a
✓The minus sign is essential. Sum is −b/a, product is c/a.
2
Writing polynomial as ax² + bx + c when 'a' is missing
✓Highest-degree coefficient MUST be non-zero. Otherwise the degree drops.
3
Saying degree of √x = 1
✓A polynomial only allows non-negative integer exponents. √x = x^(1/2) is NOT a polynomial.
4
Confusing zero of polynomial with y-intercept
✓Zero = x-value at which p(x) = 0 (x-axis crossing). y-intercept = p(0).
5
Assuming the number of zeroes always equals the degree over the reals
✓Over reals a polynomial has AT MOST 'degree' zeroes; some may be complex or repeated.
6
Applying αβ = c/a to a cubic
✓For a cubic use αβ + βγ + γα = c/a. αβγ = −d/a. Learn the three cubic relations.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the zeroes of 4x² − 4x − 3 and verify the relations.
Q2
Form the quadratic polynomial whose zeroes are 2 + √3 and 2 − √3.
Q3
If α and β are zeroes of x² − 6x + k and α² + β² = 40, find k.
Q4
Find a cubic polynomial whose zeroes are 3, −1 and 2.
Q5
Divide p(x) = x³ − 3x² + 5x − 3 by g(x) = x² − 2. Find q(x) and r(x).
Q6
State the degree and number of zeroes of a polynomial whose graph cuts the x-axis at exactly 2 points.

📝 Notes

Polynomials

An algebraic expression like p(x) = a₀ + a₁x + a₂x² + … is a polynomial when the powers of x are non-negative whole numbers.

Types by degree

Constant (degree 0): p(x) = 5
Linear (degree 1): ax + b, one zero, graph is a line
Quadratic (degree 2): ax² + bx + c, up to 2 zeroes, graph is a parabola
Cubic (degree 3): ax³ + bx² + cx + d, up to 3 zeroes

Zeroes on the graph

Zeroes of p(x) are exactly the x-coordinates where the graph of y = p(x) crosses (or touches) the x-axis. A repeated zero shows up as the graph touching but not crossing.

Sum and product formulas — one-look table

| Polynomial | Sum of zeroes | Product of zeroes | |---|---|---| | ax² + bx + c | −b/a | c/a | | ax³ + bx² + cx + d | −b/a | −d/a | | Cubic middle term | αβ + βγ + γα = c/a | — |