Chapter 8 of 19

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Andhra Pradesh (BIEAP)Class 10 Mathematics★ AP Board SSC Mathematics · 4 marks · Paper: 1-mark MCQ + 2-mark zeroes question + 4-mark division algorithm

Polynomials

POLYNOMIAL types, zeroes of a polynomial, relationship between zeroes and coefficients of quadratic and cubic polynomials, and division algorithm. AP Board Class 10 Mathematics.

⏱ 10 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Identify degree, leading coefficient, and type of polynomial (linear, quadratic, cubic)
2Find zeroes of a quadratic polynomial by factorisation
3Apply: sum of zeroes = −b/a, product = c/a for quadratic ax² + bx + c
4Apply cubic polynomial zero relationships: sum = −b/a, pairwise sum = c/a, product = −d/a
5Form a quadratic polynomial given its zeroes
6Apply the Division Algorithm: p(x) = g(x)·q(x) + r(x)

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Why this chapter matters

Polynomials introduces the structural relationship between a polynomial's ROOTS and its COEFFICIENTS — a theme that returns in quadratics, cubic equations, and EAMCET. The Division Algorithm is tested in 4-mark questions. This chapter rewards 2 hours of focused practice with reliable marks.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Algebraic expressions and factorisation (Class 8-9)

Splitting the middle term — needed to find zeroes

Polynomials

Definition: p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ (aₙ≠0). Degree = n.

Types: Linear (degree 1). Quadratic (degree 2). Cubic (degree 3).

Zero of a Polynomial: Value of x for which p(x) = 0.

Relationship Between Zeroes and Coefficients

Quadratic ax²+bx+c: Sum of zeroes = −b/a. Product = c/a.

Cubic ax³+bx²+cx+d: Sum = −b/a. Sum of products taken two at a time = c/a. Product = −d/a.

Division Algorithm: p(x) = g(x)·q(x) + r(x). Degree of r(x) < degree of g(x).

Finding Zeroes Graphically — Where the graph crosses the x-axis.

Geometric Meaning of Zeroes

Linear Polynomial p(x) = ax + b

The graph of a linear polynomial is a STRAIGHT LINE. It crosses the x-axis at EXACTLY ONE point. The x-coordinate of this point is the zero of the polynomial.

Example: p(x) = 2x − 6. The graph is a line through (0,−6) and (3,0). It crosses the x-axis at x = 3. So x = 3 is the zero. Check: p(3) = 2(3)−6 = 0 ✓.

'For a linear polynomial p(x) = ax+b, the zero is ALWAYS at x = −b/a.'

Quadratic Polynomial p(x) = ax² + bx + c

The graph is a PARABOLA. It can cross the x-axis at 0, 1, or 2 points.

Zeroes	Graph	Discriminant
2 distinct real zeroes	Parabola crosses x-axis at two points	Δ > 0
1 real zero (repeated)	Parabola TOUCHES x-axis at one point	Δ = 0
No real zeroes	Parabola does NOT touch x-axis	Δ < 0

Example: p(x) = x² − 5x + 6. The parabola crosses x-axis at x = 2 and x = 3. These are the zeroes.

Example: p(x) = x² − 4x + 4 = (x−2)². The parabola touches the x-axis only at x = 2 (repeated zero).

Example: p(x) = x² + x + 1. The parabola stays entirely ABOVE the x-axis (a > 0, Δ < 0). No real zeroes.

Relationship Between Zeroes and Coefficients — Expanded

Quadratic Polynomial ax² + bx + c

If α and β are the zeroes:

Sum: α + β = −b/a
Product: αβ = c/a

Example 1: Find the zeroes of x² − 5x + 6 and verify the relationship. x² − 5x + 6 = (x−2)(x−3). Zeroes: α = 2, β = 3. Sum = 2+3 = 5. −b/a = −(−5)/1 = 5 ✓. Product = 2×3 = 6. c/a = 6/1 = 6 ✓.

Example 2: Find a quadratic polynomial with zeroes 4 and −3. Sum = 4+(−3) = 1. Product = 4×(−3) = −12. Polynomial = x² − (sum)x + product = x² − x − 12. Verify: x² − x − 12 = (x−4)(x+3). Zeroes are 4 and −3 ✓.

Example 3: If α and β are zeroes of 2x² − 5x + 3, find α² + β². α+β = 5/2, αβ = 3/2. α² + β² = (α+β)² − 2αβ = (25/4) − 2(3/2) = 25/4 − 3 = (25−12)/4 = 13/4.

Cubic Polynomial ax³ + bx² + cx + d

If α, β, γ are the zeroes:

Sum: α+β+γ = −b/a
Sum of products taken two at a time: αβ+βγ+γα = c/a
Product: αβγ = −d/a

Example 4: Zeroes of x³ − 6x² + 11x − 6 are 1, 2, 3. Verify. Sum = 1+2+3 = 6. −b/a = −(−6)/1 = 6 ✓. αβ+βγ+γα = 1×2+2×3+3×1 = 2+6+3 = 11. c/a = 11/1 = 11 ✓. αβγ = 1×2×3 = 6. −d/a = −(−6)/1 = 6 ✓.

Division Algorithm

For polynomials p(x) and g(x) (g(x) ≠ 0), there exist unique q(x) and r(x) such that: p(x) = g(x) · q(x) + r(x), where degree of r(x) < degree of g(x).

'This is analogous to: Dividend = Divisor × Quotient + Remainder.'

Example 5: Divide x³ − 3x² + 5x − 3 by x² − 2. Find q(x) and r(x).

          x − 3
    ──────────────
x²−2 ) x³ − 3x² + 5x − 3
        x³ − 2x
        ─────────
            −3x² + 7x − 3
            −3x² + 6
            ────────────
                 7x − 9

q(x) = x − 3, r(x) = 7x − 9. Check: (x²−2)(x−3) + (7x−9) = x³−3x²−2x+6+7x−9 = x³−3x²+5x−3 ✓.

Example 6: On dividing x³ − 3x² + x + 2 by g(x), the quotient is x² − x + 1 and remainder is −2x + 4. Find g(x). p(x) = g(x)·q(x) + r(x) g(x) = [p(x) − r(x)]/q(x) = [(x³−3x²+x+2) − (−2x+4)] / (x²−x+1) = (x³−3x²+3x−2) / (x²−x+1) = x − 2. (Perform polynomial division.) So g(x) = x − 2.

AP Exam Focus

Topic	Marks	Frequency
Relationship between zeroes and coefficients	3-4	Very Common
Finding quadratic polynomial from zeroes	2-3	Very Common
Division algorithm	3-4	Common
Zeroes of cubic polynomial	4	Moderate
Graphical meaning of zeroes	2-3	Moderate

Self-Test Questions

Find the zeroes of x² − 2x − 8. (Answer: (x−4)(x+2) → zeroes: 4, −2)
Find a quadratic polynomial whose zeroes are 3 and −5. (Answer: x² + 2x − 15)
If α and β are zeroes of 3x² + 4x + 1, find 1/α + 1/β. (Answer: α+β=−4/3, αβ=1/3. 1/α+1/β = (α+β)/αβ = −4)
Divide x³ + 2x² − 5x − 6 by x + 1. Find q(x) and r(x). (Answer: q(x)=x²+x−6, r(x)=0)
Verify that 1, −2, 3 are zeroes of x³ − 2x² − 5x + 6. Check the relationship between coefficients and zeroes.
Find the zeroes of 4x² − 4x − 3 and verify the relationship. (Answer: zeroes: 3/2, −1/2. Sum=1=−(−4)/4=1 ✓, Product=−3/4=c/a ✓)

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Zero of a polynomial

p(α) = 0 defines the zero α

Graphically: the x-coordinate where the parabola (or line) CROSSES (or touches) the x-axis.

Quadratic — sum of zeroes

α + β = −b/a

For ax² + bx + c. The NEGATIVE sign is critical — often dropped by mistake.

Quadratic — product of zeroes

αβ = c/a

No sign change. Product of zeroes = c/a directly.

Form quadratic from zeroes

x² − (α+β)x + αβ = 0

Or: x² − (sum)x + (product). Then scale if needed.

Cubic — sum of zeroes

α + β + γ = −b/a

For ax³ + bx² + cx + d.

Cubic — pairwise product

αβ + βγ + γα = c/a

Sum of products taken two at a time.

Cubic — triple product

αβγ = −d/a

Product of ALL three zeroes = −d/a.

Division Algorithm for polynomials

p(x) = g(x) · q(x) + r(x), where deg(r) < deg(g)

Direct analogue of integer division: Dividend = Divisor × Quotient + Remainder.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Writing sum of zeroes as b/a instead of −b/a

✓ Sum = NEGATIVE b over a. Memory aid: 'Sum is Shy (negative).' Product = c/a with no sign change.

WATCH OUT

✗ Forgetting to NEGATE when forming the quadratic: writing x² + (sum)x + product

✓ Correct form is x² − (sum)x + (product). The MINUS before the sum bracket is mandatory.

WATCH OUT

✗ In the division algorithm, stopping before the remainder has a lower degree than the divisor

✓ Continue dividing until the degree of the remainder is STRICTLY LESS than the degree of the divisor. Then STOP.

WATCH OUT

✗ For cubic: writing product of zeroes as d/a

✓ Product of three zeroes = −d/a (negative d over a). The sign flips for the triple product but not for the pairwise sum.

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Polynomials?

6 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

6 questions~4 min worth ~4 marks in Andhra Pradesh (BIEAP) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Degree of polynomial = highest power of x. Linear (1), Quadratic (2), Cubic (3).
•Zero of p(x): value of x where p(x) = 0. Graphically, where the graph CROSSES (or touches) the x-axis.
•Quadratic ax² + bx + c: sum of zeroes = −b/a (NEGATIVE), product = c/a (no sign change).
•Cubic ax³ + bx² + cx + d: sum = −b/a, pairwise sum = c/a, triple product = −d/a.
•Form quadratic from zeroes α, β: x² − (α+β)x + αβ.
•Division Algorithm: Dividend = Divisor × Quotient + Remainder. Degree of remainder < degree of divisor.
•A polynomial of degree n has at most n zeroes.
•Graphical: linear polynomial has exactly 1 zero. Quadratic has 0, 1, or 2 zeroes (discriminant).
•α² + β² = (α+β)² − 2αβ — key identity for derived quantities.
•Verify division: always substitute back to check p(x) = g(x)q(x) + r(x).

Andhra Pradesh (BIEAP) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 4 marks

Question type	Marks each	Typical count	What it tests
Very Short Answer (VSA)	1	2	Definition, formula recall
Short Answer (SA)	2	2	Direct application
Long Answer (LA)	4	1	Multi-step problem with full working

Prep strategy

AP SSC board rewards clearly shown working — never skip steps
Memorise all formulas: AP board MCQs test formula recall directly
Practise 4-mark questions with full working under timed conditions
For word problems: identify variables, write equation, solve, state answer with units

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Signal processing

Polynomials model filters and waveforms. The zeroes of a transfer function determine stability of electronic systems.

Computer graphics — Bézier curves

Smooth curves in graphics software (Photoshop, Illustrator) are cubic polynomial splines.

Physics — projectile motion

Height h(t) = −5t² + 30t + 2 is a quadratic polynomial. Setting h = 0 finds when the ball lands.

Cryptography

Polynomial arithmetic over finite fields is the basis of error-correcting codes (QR codes, DVDs).

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For zeroes and coefficients: ALWAYS write a, b, c explicitly before substituting. Avoids sign errors.
Quick check after finding zeroes: verify sum = −b/a and product = c/a before writing the final answer.
For division algorithm: carry 0 placeholders if a degree term is missing (e.g., x³ + 0x² + 5x − 3). Prevents alignment errors.
For 'find third zero' problems: use the SUM formula (easiest). Verify with PRODUCT formula (3 seconds extra, earns full marks).
4-mark division problems: always VERIFY at the end — write 'Verification: g(x)q(x) + r(x) = p(x) ✓'. Examiners reward this.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Vieta's formulas generalise: for any degree-n polynomial, the k-th elementary symmetric polynomial of the roots equals (−1)ᵏ × (coefficient of xⁿ⁻ᵏ) / (leading coefficient).
Factor Theorem (Class 10 extension): p(a) = 0 iff (x − a) is a factor of p(x). Used to check divisibility.
Newton's identities relate power sums (αⁿ + βⁿ + γⁿ) to elementary symmetric polynomials — used in olympiad problems.
A polynomial of degree n over the reals can have at most n real roots but EXACTLY n complex roots (Fundamental Theorem of Algebra).

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

AP Board SSC2-4 marks — zeroes/coefficients and division algorithm are standard questions

EAMCETHigh — zeroes, Vieta's formulas, and polynomial factorisation appear in algebra sections

JEE FoundationPolynomial roots and coefficients; complex roots of unity

IMO/INMOPolynomial inequalities and root bounding problems

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

For ax² + bx + c = a(x − α)(x − β) = a[x² − (α+β)x + αβ]. Comparing: b/a = −(α+β), so α+β = −b/a. The negative sign comes from the factored form.

Yes! A quadratic like x² + 1 has no real zeroes (discriminant < 0). Its graph is entirely above the x-axis. But it has two COMPLEX zeroes (i and −i).

If r(x) = 0, then g(x) divides p(x) exactly. This means the zeroes of g(x) are also zeroes of p(x) — g(x) is a FACTOR of p(x).

If zeroes are α, β, γ: p(x) = x³ − (α+β+γ)x² + (αβ+βγ+γα)x − αβγ. This is just the reverse of the zero-coefficient relationships.

Yes! A polynomial's definition depends on its degree (highest power), not on whether it has real zeroes. x² + 1 is quadratic (degree 2) with no real zeroes.

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