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Algebra 2 Polynomial functions

Polynomial long division and synthetic division

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Polynomial Long and Synthetic Division

Common Core Algebra 2 • Standard A-APR.6 • Polynomial Functions

Polynomial Long and Synthetic Division is a topic in Polynomial Functions in the Common Core State Standards. It is aligned to Standard A-APR.6, which requires students to rewrite rational expressions using inspection, long division, or a computer algebra system.

Polynomial division gives a quotient and remainder by long division (any divisor) or synthetic division (divisor \(x-c\)).

Common Core Algebra 2 › Polynomial Functions › Polynomial Long and Synthetic Division  —  Standard A-APR.6

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Theory

Polynomial division writes a dividend as divisor times quotient plus a remainder:

\[\text{dividend}=\text{divisor}\cdot\text{quotient}+\text{remainder}.\]
  • Long division works for any divisor.
  • Synthetic division is a shortcut only when the divisor is \(x-c\).
A remainder of \(0\) means the divisor is a factor.
Synthetic division Synthetic division of a cubic by x minus 3 gives the quotient and remainder. 3 1 -2 0 -4 3 3 9 1 1 3 5 quotient x²+x+3, remainder 5
Synthetic division by \(x-3\): quotient \(x^2+x+3\), remainder \(5\).
Division facts Division facts Division facts dividend = divisor · quotient + remainder synthetic: only for divisor x - c long division: any divisor
Long vs synthetic division.

The division identity:

\[P(x)=(x-c)\,Q(x)+r\]
a polynomial equals the divisor times the quotient plus the remainder
Include zero coefficients for missing powers.

How to divide

  1. Write coefficients in order, using \(0\) for missing powers.
  2. For synthetic, bring down, multiply by \(c\), add — repeat.
  3. For long division, divide leading terms, multiply, subtract, repeat.
  4. Read off the quotient and remainder.
Example 1 — Synthetic division
Divide \(x^3-2x^2-4\) by \(x-3\).
Solution

Use root \(3\) with coefficients \(1,-2,0,-4\).

\(\text{quotient}\)\(=\)\(x^2+x+3\)
\(\text{remainder}\)\(=\)\(5\)
quotient x squared plus x plus 3, remainder 5
Example 2 — Long division
Divide \(2x^2+3x-5\) by \(x+2\).
Solution

Divide leading terms, multiply, subtract, repeat.

\(2x^2+3x-5\)\(=\)\((x+2)(2x-1)-3\)
\(\text{quotient}\)\(=\)\(2x-1,\ \text{remainder } -3\)
quotient 2 x minus 1, remainder negative 3
Example 3 — Exact division
Divide \(x^2-x-6\) by \(x-3\).
Solution

Root \(3\), coefficients \(1,-1,-6\).

\(\text{quotient}\)\(=\)\(x+2\)
\(\text{remainder}\)\(=\)\(0\)
quotient x plus 2, remainder 0, so it divides evenly
Example 4 — Write the result
Express \(\dfrac{x^2-x-6}{x-3}\) from Example 3.
Solution

A zero remainder means the division is exact.

\(\dfrac{x^2-x-6}{x-3}\)\(=\)\(x+2\)
the quotient is x plus 2 exactly

Common pitfalls

Include zeros for missing powers before dividing.
Synthetic uses \(c\), not \(-c\): for \(x-3\) the number is \(3\).
The last value is the remainder, not part of the quotient.

Frequently asked questions

When can you use synthetic division?

Only when the divisor has the form \(x-c\).

What does a remainder of zero mean?

The divisor is a factor of the polynomial.

What number goes in the synthetic box for \(x-3\)?

\(+3\) — the value of \(c\).

Why include zero coefficients?

To keep place value for missing powers so the division lines up.

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