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 . The division result is written as , where is the dividend, is the divisor, is the quotient, and is the remainder. The Remainder Theorem says that dividing by gives remainder , and the Factor Theorem says is a factor when .
Key Facts
- Polynomial division follows the identity , where the degree of must be less than the degree of .
- 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 .
- Synthetic division can be used only when the divisor is linear and written as , so the synthetic number is .
- When dividing by , the remainder equals by the Remainder Theorem.
- If , then is a factor of by the Factor Theorem.
- Missing powers must be included with zero coefficients, such as .
- The final answer may be written as when the remainder is not zero.
Vocabulary
- Dividend
- The dividend is the polynomial being divided, labeled in .
- Divisor
- The divisor is the polynomial you divide by, labeled in the division identity.
- Quotient
- The quotient is the main result of division, labeled in .
- 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 .
- Remainder Theorem
- The Remainder Theorem states that the remainder when is divided by is .
Common Mistakes to Avoid
- Using synthetic division with a divisor that is not in the form is wrong because the shortcut only matches linear divisors with leading coefficient .
- Forgetting zero placeholders is wrong because missing powers change the column alignment, such as treating as if it had no coefficient.
- Changing the wrong sign for the synthetic number is wrong because the divisor uses , so must use .
- 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 .
Practice Questions
- 1 Divide by using synthetic division.
- 2 Divide by and write the answer in the form .
- 3 Use long division to divide by .
- 4 Explain why synthetic division is not the best method for dividing by .
Understanding Polynomial Long Division & Synthetic Division
The main idea behind the division process is cancellation. At each stage, you choose a quotient term that removes the highest-power term currently left in the dividend. This is why terms must be written in descending order before any work begins.
A missing power can break the alignment of every later step. For example, a polynomial with a cubic term, then a linear term, needs a zero coefficient in the quadratic position.
Think of those zero terms as empty place holders. They keep like powers in the correct columns, much like zeros in place value keep digits lined up in whole-number subtraction.
Long division is the safer method because it shows where every expression comes from. After finding one quotient term, multiply the entire divisor by that term. Then subtract the full result from the current polynomial.
The most common error happens here. Students subtract only the first term or forget that subtraction changes every sign in the grouped expression. Writing the product in parentheses before changing signs makes this step easier to control.
Each new line should have matching powers directly under each other. If a term seems to disappear unexpectedly, check whether a zero term was needed or whether signs were changed correctly.
Synthetic division compresses the same reasoning into a table of coefficients. The number used in the table comes from rewriting the divisor in the form x minus c. A divisor written as x plus four uses negative four, since x plus four is the same as x minus negative four.
The first coefficient comes straight down. Each later entry is found by multiplying the previous result by the synthetic number, then adding it to the next coefficient. The numbers produced before the last one are coefficients of the quotient.
Their powers are one lower than the powers in the original dividend. The final number has a different job. It is the remainder, not another quotient coefficient.
Division gives useful information about roots and graphs. When a remainder is zero, the divisor identifies a value of x where the polynomial has value zero. That value is an x-intercept of the graph, unless the graph only touches the axis before turning around.
Repeated division can reveal more factors after one root has been found. Students use this process when solving cubic or fourth-degree equations, especially after testing simple possible roots. A strong final check is to multiply the divisor by the quotient, then add the remainder.
The result must rebuild the original polynomial exactly. This check catches incorrect coefficients, missing powers, and sign errors before they affect later factoring or graphing work.