Polynomial Long Division

➗ Polynomial Long Division

Part 1 of 5 — Why Divide Polynomials?

Topics in This Part

🔑 Key Concept: Polynomial long division works exactly like the long division you learned for whole numbers — you just divide, multiply, subtract, and bring down. The numbers are replaced by terms like $x^2$ and $3x$, but the rhythm is identical.

The Division Vocabulary

Every division problem has the same four roles. Compare arithmetic to polynomials:

Concept Check 🎯

Standard Form & Placeholders

Before dividing, always write both polynomials in standard form (highest power first) and fill every gap with a zero coefficient.

If a power is missing, insert it as $0 \cdot x^n$. This keeps your columns lined up — just like writing $204$ instead of "$2\;\;4$" so the place values don't shift.

Example

Standard Form & Placeholders 🔽

Choose the correct full standard form for each polynomial.

One More Idea: Degree

The degree of a polynomial is its highest exponent — $x^4 - 5$ has degree $4$, and $x - 1$ has degree $1$. Degree is what tells you dividing (you'll use it in every problem from here on).

Count the Placeholders 🧮

To divide $x^4 - 5$ by $x - 1$, you first rewrite the dividend with placeholders:

$x^4 + 0x^3 + 0x^2 + 0x - 5$

You're Set Up to Divide

You now know the vocabulary (dividend, divisor, quotient, remainder), the rule that the remainder is always lower degree than the divisor, and the must-do prep step of standard form + placeholders.

In Part 2 we run the full divide–multiply–subtract–bring-down loop on a real example.

➗ Polynomial Long Division

Part 2 of 5 — The Four-Step Loop

🔑 The Loop: Every step of polynomial long division repeats four moves — Divide, Multiply, Subtract, Bring down (D-M-S-B). Memorize this rhythm and the whole method becomes mechanical.

The Four Steps

To divide, you repeat this loop until the leftover has lower degree than the divisor:

1. Divide the leading term of the current dividend by the leading term of the divisor. This gives the next term of the quotient.
2. Multiply that quotient term by the entire divisor.
3. Subtract the product from the current dividend.
4. Bring down the next term and repeat.

⚠️ Subtracting flips signs. When you subtract a product, change the sign of every term you're subtracting. Distributing that minus sign is where most arithmetic errors happen.

Worked Example: $(x^2 + 5x + 6) \div (x + 2)$

➗ Polynomial Long Division

Part 3 of 5 — Remainders & Placeholders

🔑 Reality Check: Most divisions do not come out evenly. When a remainder is left over, we write the answer as $\text{quotient} + \dfrac{\text{remainder}}{\text{divisor}}$ — exactly like a mixed number.

Writing a Remainder

When the leftover has lower degree than the divisor, you stop — that leftover is the remainder $$. The complete answer is:

➗ Polynomial Long Division

Part 4 of 5 — The Remainder & Factor Theorems

🔑 Big Payoff: Division isn't just busywork. The Remainder Theorem lets you find a remainder without dividing, and the Factor Theorem turns division into a tool for factoring and finding roots.

The Remainder Theorem

🔑 Remainder Theorem: When a polynomial $P(x)$ is divided by $(x - c)$, the remainder equals $$.

➗ Polynomial Long Division

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) set up with standard form and placeholders, (2) run the divide–multiply–subtract–bring-down loop, (3) write remainders correctly, and (4) use the Remainder and Factor Theorems. Time to put it all together.

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