Is it correct to say: Writing α + β = b/a (missing negative sign)?

Sum of zeros = -b/a. The minus is essential.

Is it correct to say: Applying sum/product formulas without dividing by a?

Use α + β = -b/a and αβ = c/a. When a = 1 the division is trivial — still write it.

Is it correct to say: Using the same product sign for cubics as for quadratics?

For cubics, αβγ = -d/a (negative). For quadratics, αβ = c/a (positive).

Is it correct to say: Ignoring the leading coefficient k when forming a polynomial from zeros?

p(x) = k(x² − (α+β)x + αβ). Any k ≠ 0 works — take k = 1 for the simplest form.

Is it correct to say: Confusing degree with number of terms?

Degree = highest power of x; e.g. x² + x + 1 has degree 2, three terms.

Is it correct to say: Concluding a polynomial has 3 zeros just because it is cubic?

Real zeros can be fewer (1 or 2). The 'fundamental' count of 3 includes complex roots.

Formula sheet (PDF) →

Mathematics · Chapter 2

Polynomials

Zeros of quadratic and cubic polynomials, relations between zeros and coefficients, division algorithm — Maharashtra SSC Algebra

10 formulas · 3 solved examples · 6 practice questions · Board weight: 4-5 marks · Maharashtra State Board Class 10

Not in the 2025–26 syllabus. This chapter is not part of the Maharashtra SSC (Std X) syllabus for 2025–26. It is kept here for reference.

Board Exam Tips

→Sum and product of zeros are compulsory in every Algebra paper — form-a-polynomial is a 2-mark starter.
→For cubic, sum of zeros = -b/a, sum of products taken two at a time = c/a, product of zeros = -d/a. Memorise all three.
→Marathi terms: बहुपदी (polynomial), शून्ये (zeros), भागाकार अल्गोरिदम (division algorithm).
→Draw parabola shape: opens up if a > 0, opens down if a < 0. Number of zeros = number of x-axis crossings.
→Verify computed zeros by substituting back into p(x) — catches sign mistakes and gives 1 confirmation mark.

Formulas(10)

Standard Form of Quadratic PolynomialBoard favourite

p (x) = a x^{2} + b x + c, a \neq = 0

Symbol	Meaning
$a, b, c$	Real coefficients with a ≠ 0

Sum of Zeros (Quadratic)Board favourite

α + β = - \frac{b}{a}

Symbol	Meaning
$α, β$	Zeros (roots) of the quadratic

Product of Zeros (Quadratic)Board favourite

α β = \frac{c}{a}

Quadratic Polynomial from Given Zeros

p (x) = k [x^{2} - (α + β) x + α β]

Sum of Zeros (Cubic)

α + β + γ = - \frac{b}{a}

Sum of Products Taken Two at a Time (Cubic)

α β + β γ + γ α = \frac{c}{a}

Product of Zeros (Cubic)

α β γ = - \frac{d}{a}

Division Algorithm for Polynomials

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

Remainder Theorem

p (x) divided by (x - a) \Rightarrow r = p (a)

Factor Theorem

(x - a) is a factor ⟺ p (a) = 0

Solved Examples

1Solved Exampleeasy4 steps

If the zeros of the polynomial p(x) = x² − 3x − 10 are α and β, verify the relations between zeros and coefficients.

Factorise

x^{2} - 3 x - 10 = (x - 5) (x + 2)

2Solved Exampleboard4 steps

Find a quadratic polynomial whose zeros are 3 + √2 and 3 − √2.

Sum of zeros

α + β = (3 + 2) + (3 - 2) = 6

3Solved ExampleHOTS5 steps

If two zeros of the polynomial p(x) = x³ − 4x² − 7x + 10 are −2 and 1, find the third zero.

Compare with ax³ + bx² + cx + d

a = 1, b = - 4, c = - 7, d = 10

Traps & Common Mistakes

Common Mistakes6

1
Writing α + β = b/a (missing negative sign)
✓Sum of zeros = -b/a. The minus is essential.
2
Applying sum/product formulas without dividing by a
✓Use α + β = -b/a and αβ = c/a. When a = 1 the division is trivial — still write it.
3
Using the same product sign for cubics as for quadratics
✓For cubics, αβγ = -d/a (negative). For quadratics, αβ = c/a (positive).
4
Ignoring the leading coefficient k when forming a polynomial from zeros
✓p(x) = k(x² − (α+β)x + αβ). Any k ≠ 0 works — take k = 1 for the simplest form.
5
Confusing degree with number of terms
✓Degree = highest power of x; e.g. x² + x + 1 has degree 2, three terms.
6
Concluding a polynomial has 3 zeros just because it is cubic
✓Real zeros can be fewer (1 or 2). The 'fundamental' count of 3 includes complex roots.

Practice Yourself

Practice Yourself6 questions

Q1
Find the zeros of p(x) = x² − 2x − 8 and verify sum/product of zeros.
Q2
Form the polynomial whose zeros are -3 and 4.
Q3
Divide p(x) = x³ − 3x² + 5x − 3 by g(x) = x − 1 and find quotient and remainder.
Q4
If α, β are zeros of x² + 5x + 6, find (i) 1/α + 1/β, (ii) α² + β².
Q5
Find k so that x = 2 is a zero of p(x) = x³ − kx² + 3x − 4.
Q6
If α and β are zeros of x² − 6x + 8, form the polynomial whose zeros are 2α and 2β.

Notes

Polynomials

Polynomials sit at the crossroads of algebra and coordinate geometry. Their zeros tell us where the curve meets the x-axis, and the coefficients encode the sum, product and combinations of those zeros — long before we actually solve the equation.

SSC angle

Maharashtra Board Algebra places one direct 2-mark and one 3-mark polynomial question every year:

2-mark: verify relations between zeros and coefficients.
3-mark: form a polynomial from given zeros, or find missing zero of a cubic.
HOTS: apply division algorithm to find quotient and remainder, or use factor theorem.

Choose the right form

When zeros are known → p(x) = x² − (sum)x + (product).
When one factor is known → use division algorithm.
When one zero is given → factor out (x − a), then solve the resulting quadratic.

Handy identities that save time

α² + β² = (α + β)² − 2αβ
(α − β)² = (α + β)² − 4αβ
α³ + β³ = (α + β)³ − 3αβ(α + β)
1/α + 1/β = (α + β)/(αβ)

Number of real zeros vs degree

Linear (degree 1): 1 real zero.
Quadratic (degree 2): 0, 1 (repeated) or 2 real zeros — count using discriminant.
Cubic (degree 3): at least 1 real zero; up to 3.

Board vs CBSE

Both boards teach the same content. Maharashtra SSC leans on Vieta-style questions (sum/product manipulations) and uses the term "बहुपदी" freely in Marathi-medium paper.