Lesson 5.1

Polynomial Long Division

The same algorithm you learned in 4th grade, now supersized for algebra. Break down complex functions into simpler parts.

Introduction

The same algorithm you learned in 4th grade, now supersized for algebra. Break down complex functions into simpler parts.

Past Knowledge

Remember doing ? You got 5 with a remainder of 2. We wrote it as .

Today's Goal

We apply this exact logic to polynomials. Dividing by quotient gives us a quotient and a remainder .

Future Success

You cannot integrate a fraction like directly. But if you divide it first, it becomes , which is easy to integrate!

Key Concepts

The Division Algorithm

Dividend
Divisor
=
Quotient +
Remainder
Divisor
1. Divide
Divide the leading term of the dividend by the leading term of the divisor.
2. Multiply
Multiply the result by the ENTIRE divisor.
3. Subtract
Subtract your result from the dividend. Be careful with signs!
4. Repeat
Bring down the next term and do it all again.

Worked Examples

Level: Basic

Example 1: Dividing by a Linear Binomial

Divide .

x + 3
x + 2
x² + 5x + 6
-(x² + 2x)
3x + 6
-(3x + 6)
0
Result
Quotient: , Remainder: 0.
Level: Intermediate

Example 2: The "Missing Term" Trap

Divide .

Placeholder Alert! The dividend is missing and terms. You MUST write them as and to keep columns aligned.
x² + x + 1
x - 1
x³ + 0x² + 0x - 1
-(x³ - x²)
x² + 0x
-(x² - x)
x - 1
-(x - 1)
0
Result after dividing: .
Since Remainder is 0, we can say .
Level: Advanced

Example 3: Interpreting the Remainder

Divide by .

2x + 1
x + 1
2x² + 3x + 5
-(2x² + 2x)
x + 5
-(x + 1)
4
Final Answer Form

Common Pitfalls

  • Sign Errors on Subtraction:

    When you subtract the row, you must distribute the negative sign to BOTH terms. It helps to physically write the sign change in a different color circle on your paper.

  • Stopping Too Early:

    You are only done when the degree of the remainder is strictly LESS than the degree of the divisor. If you have an x term left and are dividing by x, keep going!

Real-Life Applications

Digital Communication: Cyclic Redundancy Checks (CRC)

When your computer sends data over Wi-Fi, it treats the data bits (101101...) as coefficients of a massive polynomial.

To check for errors, it divides this polynomial by a specialized "generator polynomial" and sends the remainder along with the message. The receiver divides the message by the same generator. If they get a different remainder, they know the data was corrupted and ask for it again.

Practice Quiz

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