Polynomial long division and synthetic division

Overview

Dividing polynomials is like dividing numbers, but we track powers of $x$ instead of place values. Two methods exist: long division (works for any divisor) and synthetic division (faster, but only for linear divisors of the form $x - c$).

Why learn this? Factoring polynomials, finding roots, simplifying rational expressions, and understanding the Remainder Theorem all depend on polynomial division.

[!intuition] The Core Idea

When you divide $847 \div 23$, you ask: "How many 23s fit into 84?" Then subtract and bring down the next digit. Polynomial division is identical, except:

• We ask "How many times does the leading term of the divisor fit into the leading term of the dividend?"
• We subtract entire polynomial expressions
• We "bring down" the next term

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

For polynomials: $P(x) = D(x) \cdot Q(x) + R(x) \text{ where } \deg(R) < \deg(D) \text{ or } R = 0.$

with $\deg(R(x)) < \deg(D(x))$ or $R(x) = 0$.

WHY does this work? Each step eliminates the highest-degree term of the current dividend by choosing the right quotient term. We repeatedly subtract multiples of $D(x)$ until what remains has degree less than $D(x)$.

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

Setup: Write $P(x)$ and $D(x)$ in descending powers, filling in $0$ coefficients for missing terms.

Steps:

1. Divide leading terms: $\frac{\text{leading term of dividend}}{\text{leading term of divisor}}$ → first term of $Q(x)$
   • WHY? This tells us "how many divisors fit" at the highest power
2. Multiply: Multiply entire $D(x)$ by this quotient term
   • WHY? We're subtracting out that portion of the dividend
3. Subtract: Subtract the product from $P(x)$
   • WHY? Removes the highest-degree term, revealing the next dividend
4. Bring down: Move to the next term
5. Repeat until the degree of the remainder is less than $\deg(D(x))$

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

Setup:

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

Step 1: Divide leading terms: $\frac{2x^3}{x} = 2x^2$

• Why? We need $2x^2$ copies of $(x+2)$ to cancel the $2x^3$ term

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

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

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

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

         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: Expand $(x+2)(2x^2 - x - 3) + 7$:

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

[!definition] Synthetic Division

A shortcut for dividing by linear divisors of the form $(x - c)$. We work only with coefficients, using the value $c$ (not $x - c$!).

WHY $c$ not $-c$? If dividing by $(x - c)$, we use $+c$ in synthetic division because the algorithm implicitly adds rather than subtracts.

Restriction: Only works when divisor is first-degree with leading coefficient 1.

[!formula] Synthetic Division Algorithm

Given: $P(x) \div (x - c)$

Steps:

1. Write coefficients of $P(x)$ in descending order (include 0s)
2. Write $c$ to the left (the value that makes $x - c = 0$)
3. Bring down the first coefficient
4. Multiply by $c$, write result under next coefficient
5. Add the column
6. Repeat multiply-add until done
7. Last number is the remainder; others are quotient coefficients (one degree lower)

WHY does this work? Synthetic division is a compressed form of long division where we've removed the variable $x$ and optimized the arithmetic. Each multiply-add step mimics "multiply divisor by quotient term, then subtract."

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

Setup: Divisor is $x + 2 = x - (-2)$, so $c = -2$

Coefficients of $2x^3 + 3x^2 - 5x + 1$: $[2, 3, -5, 1]$

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

Step-by-step:

1. Bring down $2$
   • Why? First coefficient of quotient is always first coefficient of dividend
2. Multiply: $2 \times (-2) = -4$, write under $3$
   • Why? We're computing $2 \times (-2)$ — multiplying the running quotient value by $c$, not by the whole divisor
3. Add: $3 + (-4) = -1$
4. Multiply: $-1 \times (-2) = 2$, write under $-5$
5. Add: $-5 + 2 = -3$
6. Multiply: $-3 \times (-2) = 6$, write under $1$
7. Add: $1 + 6 = 7$ (remainder)

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

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

Same answer as long division!

[!example] Example 3: Divide $x^4 - 3x^2 + 2$ by $x - 1$

Note the missing $x^3$ and $x$ terms — we use 0 coefficients.

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 $0$

WHY remainder = 0? This means $x = 1$ is a root of $x^4 - 3x^2 + 2$! (Remainder Theorem)

[!mistake] Common Errors

Mistake 1: Forgetting to include 0 coefficients

Wrong approach: Divide $x^3 + 5$ by $x - 2$ using coefficients $[1, 5]$

Why it feels right: "Those terms aren't there, so skip them."

The fix: Always write $x^3 + 0x^2 + 0x + 5$ → coefficients $[1, 0, 0, 5]$ (one zero for the missing $x^2$ term, one for the missing $x$ term)

WHY? Place value matters! Just like writing $1005$ not $15$ when dividing numbers.

Mistake 2: Using wrong sign in synthetic division

Wrong: Dividing by $x + 3$, student uses $+3$ in synthetic division

Why it feels right: "The divisor has $+3$, so I use $+3$."

The fix: $x + 3 = x - (-3)$, so use $c = -3$

WHY? Synthetic division uses the zero of the divisor. $x + 3 = 0$ when $x = -3$.

Mistake 3: Misaligning terms during long division

Wrong: After subtraction, writing remainder terms in wrong columns

Why it feels right: "I just write the numbers down as I get them."

The fix: Keep powers aligned vertically — $x^2$ terms under $x^2$ terms, etc.

WHY? Polynomial arithmetic requires combining like terms. Misalignment = adding $x^2$ to $x$.

Mistake 4: Stopping too early

Wrong: Remainder is $-3x + 5$ when dividing by $x^2 + 1$, but student continues dividing

Why it feels right: "There's still stuff left, so keep going."

The fix: Stop when $\deg(\text{remainder}) < \deg(\text{divisor})$

WHY? A remainder of degree 1 cannot contain a divisor of degree 2. You're done!

[!recall]- Explain to a 12-year-old

Imagine you have 847 candies and want to divide them into groups of 23. You'd ask "How many 23s fit into 84?" (about 3), then multiply $3 \times 23 = 69$, subtract $84 - 69 = 15$, bring down the 7 to get 157, and keep going.

Polynomial division is exactly the same game, but instead of candies, you're dividing $x^3$ and $x^2$ terms. You ask "How many $x$s fit into $2x^3$?" (answer: $2x^2$), then multiply and subtract just like with numbers.

Synthetic division is like a cheat code — when you're dividing by something simple like $(x - 5)$, you can skip all the $x$ symbols and just work with the numbers, following a pattern of multiply-then-add. It's way faster but only works for those simple divisors!

[!mnemonic] Memory Aid

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

Synthetic Sign Rule: "Divisor $(x - c)$ uses $+c$; divisor $(x + c)$ uses $-c$" → "Flip the sign!"

Remainder Check: "If remainder degree $\geq$ divisor degree, you're not done yet!"

Connections

• 2.1.1-Polynomials-definition-and-degree — understanding polynomial structure
• 2.1.13-Factor-theorem-and-remainder-theorem — remainder connects to $P(c)$
• 2.15-Rational-root-theorem — finding candidates for $c$ in synthetic division
• 2.1.8-Factoring-polynomials — division helps factor by finding quotients
• 3.2.4-Partial-fraction-decomposition — uses polynomial division when numerator degree $\geq$ denominator degree

#flashcards/maths

What is the relationship between dividend, divisor, quotient, and remainder in polynomial division? :: $P(x) = D(x) \cdot Q(x) + R(x)$ where $\deg(R) < \deg(D)$ or $R = 0$

Concept Map

Hinglish (regional understanding)