Polynomials | Class 10 Maths

🔤 Polynomial Kya Hota Hai?

💡 Simple samajho:

Area formula: l × b. Ek circle ki circumference: 2πr. Ek ball ki height: −5t² + 20t + 1. Ye sab polynomials hain — ek ya zyada terms ka sum, jahan variables ke positive integer powers ho.

Polynomial: p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Jahan aₙ ≠ 0, aur n ek non-negative integer hai.
Degree: Sabse bada power. aₙ = leading coefficient. a₀ = constant term.

📊 Polynomial ke Types

Linear Polynomial

Degree = 1. p(x) = ax + b. Example: 3x + 5, 2x − 7.

Quadratic Polynomial

Degree = 2. p(x) = ax² + bx + c (a≠0). Example: x² − 5x + 6, 2x² + 3x − 2.

Cubic Polynomial

Degree = 3. p(x) = ax³ + bx² + cx + d. Example: x³ − 3x² + 2x − 1.

📌 Zeros of a Polynomial (Shunyas)

Zero: Woh value of x jis par p(x) = 0 ho, use polynomial ka zero kehte hain.
Geometrically — yeh woh points hain jahan graph x-axis ko cross ya touch karta hai!

Example: p(x) = x² − 5x + 6 ke zeros nikalo

x² − 5x + 6 = 0
Factorise: (x−2)(x−3) = 0
x = 2 ya x = 3
α = 2 β = 3

Verify: p(2) = ?

p(2) = 4 − 10 + 6 = 0 ✓
p(3) = 9 − 15 + 6 = 0 ✓

🔗 Zeros aur Coefficients ka Relation

Agar quadratic polynomial ax² + bx + c ke zeros α aur β hain:

Sum of Zeros

α + β = −b/a

"coefficient of x" / "coefficient of x²" ka negative

Product of Zeros

αβ = c/a

"constant term" / "coefficient of x²"

Cubic Polynomial (ax³+bx²+cx+d) ke liye

α+β+γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a

Example: x² − 5x + 6. Sum aur Product verify karo.

a=1, b=−5, c=6. Zeros: α=2, β=3
α + β = 2+3 = 5 = −(−5)/1 = −b/a ✓
αβ = 2×3 = 6 = 6/1 = c/a ✓

Example: 4s² − 4s + 1. Zeros aur verify karo.

4s² − 4s + 1 = (2s−1)² = 0 → s = 1/2 (double root)
α = β = 1/2
Sum = 1 = −(−4)/4 = 1 ✓
Product = 1/4 = 1/4 ✓

🏗️ Polynomial Banana — Zeros se

Agar zeros α aur β diye hain, polynomial bana sakte hain:
p(x) = k[x² − (α+β)x + αβ] (k ≠ 0, koi bhi constant)

Example: Zeros hain 3 aur −4. Polynomial banao.

Sum = 3 + (−4) = −1
Product = 3 × (−4) = −12
p(x) = k[x² − (−1)x + (−12)] = k(x² + x − 12)
Simplest: x² + x − 12

Example: Zeros hain 1/2 aur 3/2. Polynomial banao (integer coefficients).

Sum = 1/2 + 3/2 = 2
Product = 1/2 × 3/2 = 3/4
p(x) = k(x² − 2x + 3/4). k=4 lene par: 4x² − 8x + 3

📋 Division Algorithm for Polynomials

Theorem: Koi bhi polynomial p(x) ko non-zero polynomial g(x) se divide karne par:
p(x) = g(x) × q(x) + r(x)
jahan r(x) = 0 ya deg r(x) < deg g(x)
(Bilkul waise jaise integers mein a = bq + r)

Example: p(x) = x³ − 3x² + 5x − 3 ko g(x) = x² − 2 se divide karo

Long division karne par:
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 ✓

Important Use: Agar ek zero (α) pata ho to (x−α) se divide karo — isse baaki zeros nikal sakte hain!
Example: Agar p(x) = x³ − 4x² + 5x − 2 ka ek zero x=1 hai, to (x−1) se divide karo → q(x) = x²−3x+2 = (x−1)(x−2) → zeros: 1, 1, 2

📉 Graphs aur Zeros ka Relation

Polynomial Type	Graph Shape	Max Zeros
Linear (ax+b)	Seedhi line, ek baar cross karti hai	1
Quadratic (ax²+bx+c)	Parabola	2
Cubic (ax³+...)	S-curve	3

Parabola cases:
• D > 0 (b²−4ac > 0): Graph x-axis ko 2 different points par cross karta hai → 2 real zeros
• D = 0: Graph x-axis ko ek point par touch karta hai → 1 repeated zero
• D < 0: Graph x-axis ko kabhi cross nahi karta → 0 real zeros

📝 Practice Questions

Q1. Find zeros of p(x) = x² − 2x − 8 aur coefficients se verify karo.

x²−2x−8 = (x−4)(x+2) = 0 → zeros: 4, −2
Sum = 4+(−2) = 2 = −(−2)/1 ✓ | Product = −8 = −8/1 ✓

Q2. Quadratic polynomial banao jiske zeros hain √2 aur −√2.

Sum = 0, Product = −2. p(x) = x² − 2

Q3. p(x) = 3x² + 5x − 2. Find zeros. Coefficient relation verify karo.

3x²+5x−2 = (3x−1)(x+2) = 0 → x = 1/3, −2
Sum = 1/3 − 2 = −5/3 = −5/3 ✓ | Product = 1/3×(−2) = −2/3 = −2/3 ✓

Q4. p(x) = x³ − 3x ka ek zero √3 hai. Baaki zeros nikalo.

x³ − 3x = x(x² − 3) = x(x−√3)(x+√3)
Zeros: 0, √3, −√3

Q5. On dividing x³ − 3x² + x + 2 by g(x), we get q(x) = x−2, r(x) = −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)] / (x−2)
= [x³−3x²+3x−2] / (x−2) = x²−x+1

🎮 Quadratic Explorer → Next: Linear Equations →

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