Polynomial Long Division & Synthetic Division Cheat Sheet

Polynomial long division and synthetic division are methods for dividing one polynomial by another. Students need this cheat sheet to organize each step, track signs, and connect division to factors and zeros. These skills support factoring, graphing polynomials, simplifying rational expressions, and solving higher-degree equations.

Long division works for any polynomial divisor, while synthetic division is a shortcut for divisors of the form $x - c$ . The division result is written as $P(x) = D(x)Q(x) + R(x)$ , where $P(x)$ is the dividend, $D(x)$ is the divisor, $Q(x)$ is the quotient, and $R(x)$ is the remainder. The Remainder Theorem says that dividing $P(x)$ by $x - c$ gives remainder $P(c)$ , and the Factor Theorem says $x - c$ is a factor when $P(c) = 0$ .

Key Facts

Polynomial division follows the identity $P(x) = D(x)Q(x) + R(x)$ , where the degree of $R(x)$ must be less than the degree of $D(x)$ .
For long division, divide the leading term of the current dividend by the leading term of the divisor, then multiply, subtract, and bring down the next term.
When subtracting polynomials, distribute the negative sign to every term, such as $-(3x^2 - 5x + 1) = -3x^2 + 5x - 1$ .
Synthetic division can be used only when the divisor is linear and written as $x - c$ , so the synthetic number is $c$ .
When dividing by $x - c$ , the remainder equals $P(c)$ by the Remainder Theorem.
If $P(c) = 0$ , then $x - c$ is a factor of $P(x)$ by the Factor Theorem.
Missing powers must be included with zero coefficients, such as $2x^3 + 5 = 2x^3 + 0x^2 + 0x + 5$ .
The final answer may be written as $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$ when the remainder is not zero.

Vocabulary

Dividend: The dividend is the polynomial being divided, labeled $P(x)$ in $P(x) = D(x)Q(x) + R(x)$ .
Divisor: The divisor is the polynomial you divide by, labeled $D(x)$ in the division identity.
Quotient: The quotient is the main result of division, labeled $Q(x)$ in $P(x) = D(x)Q(x) + R(x)$ .
Remainder: The remainder is what is left after division and must have degree less than the divisor.
Synthetic Division: Synthetic division is a shortcut for dividing a polynomial by a linear divisor of the form $x - c$ .
Remainder Theorem: The Remainder Theorem states that the remainder when $P(x)$ is divided by $x - c$ is $P(c)$ .

Common Mistakes to Avoid

Using synthetic division with a divisor that is not in the form $x - c$ is wrong because the shortcut only matches linear divisors with leading coefficient $1$ .
Forgetting zero placeholders is wrong because missing powers change the column alignment, such as treating $x^3 + 2x + 1$ as if it had no $x^2$ coefficient.
Changing the wrong sign for the synthetic number is wrong because the divisor $x - c$ uses $c$ , so $x + 4$ must use $c = -4$ .
Subtracting only the first term during long division is wrong because the entire product must be subtracted, so every term changes sign.
Writing a remainder with degree equal to or greater than the divisor is wrong because division is not complete until $\deg(R) < \deg(D)$ .

Practice Questions

1 Divide $2x^3 + 3x^2 - 11x - 6$ by $x + 2$ using synthetic division.
2 Divide $x^3 - 4x^2 + 0x + 7$ by $x - 3$ and write the answer in the form $Q(x) + \frac{R(x)}{D(x)}$ .
3 Use long division to divide $3x^4 - 5x^3 + 2x - 8$ by $x^2 - 2x + 1$ .
4 Explain why synthetic division is not the best method for dividing $2x^3 + x^2 - 5$ by $x^2 + 1$ .

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Polynomial Long Division & Synthetic Division Cheat Sheet

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