Polynomial Division

Long division and synthetic division

Polynomial Division

Long Division

Same process as numerical long division!

Steps:

Divide leading terms
Multiply divisor by quotient term
Subtract
Bring down next term
Repeat

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

$\begin{array}{c|cc} & x + 3 \\ \hline x + 2 & x^2 + 5x + 6 \\ & x^2 + 2x \\ \hline & 3x + 6 \\ & 3x + 6 \\ \hline & 0 \end{array}$

Result: $x + 3$ with remainder $0$

Synthetic Division

Only works when dividing by $(x - c)$

Much faster than long division!

Steps:

Write coefficients of dividend
Use $c$ from divisor $(x - c)$
Bring down first coefficient
Multiply and add repeatedly

Example: $(2x^3 - 5x^2 + 3x - 2) \div (x - 2)$

Use $c = 2$ : $\begin{array}{c|cccc} 2 & 2 & -5 & 3 & -2 \\ & & 4 & -2 & 2 \\ \hline & 2 & -1 & 1 & 0 \end{array}$

Result: $2x^2 - x + 1$ with remainder $0$

Remainder Theorem

When dividing $f(x)$ by $(x - c)$ : $\text{Remainder} = f(c)$

Division Algorithm

$f(x) = q(x) \cdot d(x) + r(x)$

Where $q$ is quotient, $d$ is divisor, $r$ is remainder

📚 Practice Problems

1Problem 1easy

❓ Question:

Use synthetic division: $(x^2 + 7x + 10) \div (x + 2)$

💡 Show Solution

Divisor is $(x + 2)$ , so use $c = -2$

Coefficients: $1, 7, 10$

$\begin{array}{c|ccc} -2 & 1 & 7 & 10 \\ & & -2 & -10 \\ \hline & 1 & 5 & 0 \end{array}$

Process:

Bring down $1$
$1 \times (-2) = -2$ , add to $7$ → $5$
$5 \times (-2) = -10$ , add to $10$ → $0$

Answer: Quotient $x + 5$ , remainder $0$

2Problem 2medium

❓ Question:

Divide: $(3x^3 - 2x^2 + x - 5) \div (x - 1)$

💡 Show Solution

Use synthetic division with $c = 1$

$\begin{array}{c|cccc} 1 & 3 & -2 & 1 & -5 \\ & & 3 & 1 & 2 \\ \hline & 3 & 1 & 2 & -3 \end{array}$

Bottom row interpretation:

Coefficients: $3, 1, 2$ → quotient $3x^2 + x + 2$
Last number: $-3$ → remainder

Answer: $3x^2 + x + 2 - \frac{3}{x - 1}$

3Problem 3medium

❓ Question:

Use the Remainder Theorem to find the remainder when $f(x) = 2x^3 - 5x + 1$ is divided by $(x - 3)$

💡 Show Solution

By the Remainder Theorem, the remainder equals $f(3)$ .

$f(3) = 2(3)^3 - 5(3) + 1$ $= 2(27) - 15 + 1$ $= 54 - 15 + 1$ $= 40$

Answer: Remainder is $40$

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