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.

Polynomials

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

📐 10 formulas✏️ 3 examples🎯 6 practice⚖️ 4-5 marks🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

💡

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 Polynomial★ Board fav

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 fav

α + β = - \frac{b}{a}

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

Product of Zeros (Quadratic)★ Board fav

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.