Polynomial long division and synthetic division

2.1.14 · Maths › Algebra — Introduction & Intermediate

Overview

Polynomials ko divide karna dividing numbers jaisa hi hai, bas hum place values ki jagah $x$ ke powers track karte hain. Do methods hain: long division (kisi bhi divisor ke liye kaam karta hai) aur synthetic division (faster, lekin sirf $x-c$ form ke linear divisors ke liye).

Ye kyun seekhna chahiye? Polynomials ko factor karna, roots dhundna, rational expressions simplify karna, aur Remainder Theorem samajhna — sab polynomial division pe depend karta hai.

[!intuition] Core Idea

Jab aap $847÷23$ karte ho, aap poochte ho: "84 mein kitne 23 fit hote hain?" Phir subtract karo aur agla digit neeche laao. Polynomial division bilkul identical hai, bas:

• Hum poochte hain "Divisor ka leading term, dividend ke leading term mein kitni baar fit hota hai?"
• Hum poore polynomial expressions subtract karte hain
• Hum agla term "bring down" karte hain

Result: $\text{Dividend}=\text{Divisor}\times \text{Quotient}+\text{Remainder}$

Polynomials ke liye: $P(x)=D(x)\cdot Q(x)+R(x)\text{ where }\mathrm{deg}(R)<\mathrm{deg}(D)\text{ or }R=0.$

jahan $\mathrm{deg}(R(x))<\mathrm{deg}(D(x))$ ya $R(x)=0$.

Ye kaam kyun karta hai? Har step mein current dividend ka highest-degree term eliminate hota hai, kyunki hum sahi quotient term choose karte hain. Hum baar baar $D(x)$ ke multiples subtract karte rehte hain jab tak bacha hua part ki degree $D(x)$ se kam na ho jaaye.

[!formula] Long Division Algorithm (Step-by-Step)

Setup: $P(x)$ aur $D(x)$ ko descending powers mein likho, aur jo terms missing hain unke liye $0$ coefficients bharo.

Steps:

1. Leading terms divide karo: $\frac{\text{leading term of dividend}}{\text{leading term of divisor}}$ → $Q(x)$ ka pehla term
   • Kyun? Ye batata hai ki highest power par "kitne divisors fit hote hain"
2. Multiply karo: Poore $D(x)$ ko is quotient term se multiply karo
   • Kyun? Hum dividend ka woh hissa subtract kar rahe hain
3. Subtract karo: Product ko $P(x)$ se subtract karo
   • Kyun? Highest-degree term hat jaata hai, aur agla dividend reveal hota hai
4. Bring down karo: Agla term le aao
5. Repeat karo jab tak remainder ki degree $\mathrm{deg}(D(x))$ se kam na ho jaaye

[!example] Example 1: $(2x^3+3x^2-5x+1)÷(x+2)$

Setup:

         _______________
x + 2 ) 2x³ + 3x² - 5x + 1

Step 1: Leading terms divide karo: $\frac{2x^3}{x}=2x^2$

• Kyun? $2x^3$ term cancel karne ke liye humein $(x+2)$ ke $2x^2$ copies chahiye

         2x²
         _______________
x + 2 ) 2x³ + 3x² - 5x + 1
        2x³ + 4x²         (multiply: 2x²(x+2))
        ────────
            -x² - 5x      (subtract)

Step 2: Divide karo: $\frac{-x^2}{x}=-x$

         2x² - x
         _______________
x + 2 ) 2x³ + 3x² - 5x + 1
        2x³ + 4x²
        ─────────
            -x² - 5x
            -x² - 2x      (multiply: -x(x+2))
            ────
                 -3x + 1  (subtract)

Step 3: Divide karo: $\frac{-3x}{x}=-3$

         2x² - x - 3
         _______________
x + 2 ) 2x³ + 3x² - 5x + 1
        2x³ + 4x²
        ─────────
            -x² - 5x
            -x² - 2x
            ────────
                 -3x + 1
                 -3x - 6  (multiply: -3(x+2))
                ────
                       7  (remainder)

Result: $2x^3+3x^2-5x+1=(x+2)(2x^2-x-3)+7$

Verification: $(x+2)(2x^2-x-3)+7$ expand karo:

• $(x+2)(2x^2)=2x^3+4x^2$
• $(x+2)(-x)=-x^2-2x$
• $(x+2)(-3)=-3x-6$
• Sum: $2x^3+4x^2-x^2-2x-3x-6+7=2x^3+3x^2-5x+1$ ✓

[!definition] Synthetic Division

$(x-c)$ form ke linear divisors se divide karne ka ek shortcut. Hum sirf coefficients ke saath kaam karte hain, aur value $c$ use karte hain (na ki $x-c$!).

$c$ kyun, $-c$ kyun nahi? Jab $(x-c)$ se divide karte hain, to synthetic division mein $+c$ use karte hain kyunki algorithm implicitly subtract ki jagah add karta hai.

Restriction: Sirf tab kaam karta hai jab divisor first-degree ho aur leading coefficient 1 ho.

[!formula] Synthetic Division Algorithm

Diya gaya: $P(x)÷(x-c)$

Steps:

1. $P(x)$ ke coefficients descending order mein likho (0s bhi include karo)
2. $c$ ko left side par likho (woh value jo $x-c=0$ banaye)
3. Pehla coefficient neeche laao
4. $c$ se multiply karo, result agle coefficient ke neeche likho
5. Column add karo
6. Jab tak khatam na ho, multiply-add repeat karo
7. Aakhri number remainder hai; baaki quotient coefficients hain (ek degree kam)

Ye kaam kyun karta hai? Synthetic division, long division ka ek compressed form hai jismein se variable $x$ hata diya gaya hai aur arithmetic optimize ki gayi hai. Har multiply-add step, "divisor ko quotient term se multiply karo, phir subtract karo" wale step ko mimic karta hai.

[!example] Example 2: $(2x^3+3x^2-5x+1)÷(x+2)$ using Synthetic Division

Setup: Divisor $x+2=x-(-2)$ hai, isliye $c=-2$

$2x^3+3x^2-5x+1$ ke coefficients: $[2,3,-5,1]$

    -2 |  2    3   -5    1
       |     -4    2    6
       |________________
         2   -1   -3    7

Step-by-step:

1. $2$ neeche laao
   • Kyun? Quotient ka pehla coefficient hamesha dividend ka pehla coefficient hota hai
2. Multiply karo: $2\times (-2)=-4$, $3$ ke neeche likho
   • Kyun? Hum $2\times (-2)$ compute kar rahe hain — running quotient value ko $c$ se multiply kar rahe hain, poore divisor se nahi
3. Add karo: $3+(-4)=-1$
4. Multiply karo: $-1\times (-2)=2$, $-5$ ke neeche likho
5. Add karo: $-5+2=-3$
6. Multiply karo: $-3\times (-2)=6$, $1$ ke neeche likho
7. Add karo: $1+6=7$ (remainder)

Result: Quotient $2x^2-x-3$ hai, remainder $7$ hai

$2x^3+3x^2-5x+1=(x+2)(2x^2-x-3)+7$

Long division jaisa hi answer!

[!example] Example 3: $x^4-3x^2+2$ ko $x-1$ se divide karo

Note karo ki missing $x^3$ aur $x$ terms hain — hum 0 coefficients use karte hain.

Long Division:

         x³ + x² - 2x - 2
         ________________
x - 1 ) x⁴ +0x³ - 3x² + 0x + 2
        x⁴ - x³
        ────
             x³ - 3x²
             x³ - x²
             ────
                -2x² + 0x
                -2x² + 2x
                ────────
                     -2x + 2
                     -2x + 2
                     ────
                           0

Synthetic Division:

    1 |  1    0   -3    0    2
      |       1    1   -2   -2
      |______________________
        1    1   -2   -2    0

Result: $x^4-3x^2+2=(x-1)(x^3+x^2-2x-2)$ with remainder $0$

Remainder = 0 kyun? Iska matlab hai $x=1$, $x^4-3x^2+2$ ka ek root hai! (Remainder Theorem)

[!mistake] Common Errors

Mistake 1: 0 coefficients include karna bhool jaana

Galat approach: $x^3+5$ ko $x-2$ se divide karo, coefficients $[1,5]$ use karo

Kyun sahi lagta hai: "Woh terms hain hi nahi, to skip karo."

Fix: Hamesha $x^3+0x^2+0x+5$ likho → coefficients $[1,0,0,5]$ (missing $x^2$ term ke liye ek zero, missing $x$ term ke liye ek)

Kyun? Place value matter karti hai! Jaise numbers divide karte waqt $1005$ likhte hain na ki $15$.

Mistake 2: Synthetic division mein galat sign use karna

Galat: $x+3$ se divide karte waqt, student synthetic division mein $+3$ use karta hai

Kyun sahi lagta hai: "Divisor mein $+3$ hai, to main $+3$ use karunga."

Fix: $x+3=x-(-3)$, isliye $c=-3$ use karo

Kyun? Synthetic division, divisor ka zero use karta hai. $x+3=0$ jab $x=-3$.

Mistake 3: Long division mein terms misalign ho jaana

Galat: Subtraction ke baad, remainder terms galat columns mein likhna

Kyun sahi lagta hai: "Main bas numbers likhta jaata hoon jaise milte hain."

Fix: Powers vertically aligned rakho — $x^2$ terms, $x^2$ terms ke neeche, etc.

Kyun? Polynomial arithmetic mein like terms combine karna zaroori hai. Misalignment matlab $x^2$ ko $x$ se add karna.

Mistake 4: Bahut jaldi ruk jaana

Galat: Remainder $-3x+5$ hai jab $x^2+1$ se divide kar rahe ho, lekin student divide karta rehta hai

Kyun sahi lagta hai: "Abhi bhi kuch bacha hai, to chalte rehna chahiye."

Fix: Jab $\mathrm{deg}(\text{remainder})<\mathrm{deg}(\text{divisor})$ ho tab ruko

Kyun? Degree 1 ka remainder, degree 2 ke divisor ko contain nahi kar sakta. Aap done ho!

[!recall]- Ek 12-saal ke bacche ko samjhao

Socho tumhare paas 847 candies hain aur unhe 23 ke groups mein divide karna hai. Tum poochte: "84 mein kitne 23 fit hote hain?" (lagbhag 3), phir $3\times 23=69$ multiply karo, $84-69=15$ subtract karo, 7 neeche laao to 157 milta hai, aur chalte rehte ho.

Polynomial division bilkul same game hai, bas candies ki jagah tum $x^3$ aur $x^2$ terms divide kar rahe ho. Tum poochte ho "Kitne $x$ fit hote hain $2x^3$ mein?" (jawab: $2x^2$), phir numbers ki tarah hi multiply aur subtract karo.

Synthetic division ek cheat code ki tarah hai — jab tum kisi simple cheez jaise $(x-5)$ se divide kar rahe ho, to tum saare $x$ symbols skip kar sakte ho aur sirf numbers ke saath kaam kar sakte ho, multiply-then-add ka pattern follow karke. Ye bahut faster hai lekin sirf unhi simple divisors ke liye kaam karta hai!

[!mnemonic] Memory Aid

"DMSB" Long Division ke liye: Divide, Multiply, Subtract, Bring down (repeat)

Synthetic Sign Rule: "Divisor $(x-c)$ mein $+c$ use karo; divisor $(x+c)$ mein $-c$ use karo" → "Sign flip karo!"

Remainder Check: "Agar remainder degree $\ge$ divisor degree hai, to abhi khatam nahi hua!"

Connections

• 2.1.1-Polynomials-definition-and-degree — polynomial structure samajhna
• 2.1.13-Factor-theorem-and-remainder-theorem — remainder, $P(c)$ se connect hota hai
• 2.15-Rational-root-theorem — synthetic division mein $c$ ke candidates dhundna
• 2.1.8-Factoring-polynomials — division, quotients dhundkar factor karne mein madad karta hai
• 3.2.4-Partial-fraction-decomposition — polynomial division use karta hai jab numerator degree $\ge$ denominator degree ho

#flashcards/maths

Polynomial division mein dividend, divisor, quotient, aur remainder ka kya relationship hai? :: $P(x)=D(x)\cdot Q(x)+R(x)$ where $\mathrm{deg}(R)<\mathrm{deg}(D)$ or $R=0$

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