Dividing by a Monomial

Divide: $\frac{12x^3 - 8x^2 + 4x}{4x}$

Set up the division: Divide each term of the numerator by $4x$ = $\frac{12x^3}{4x} - \frac{8x^2}{4x} + \frac{4x}{4x}$

Divide the first term: $\frac{12x^3}{4x} = 3x^2$ = $3x^2$

Divide the second term: $\frac{8x^2}{4x} = 2x$ = $-2x$

Divide the third term: $\frac{4x}{4x} = 1$ = $+1$

Combine results: Put all terms together = $3x^2 - 2x + 1$

Answer: $3x^2 - 2x + 1$

Polynomial Long Division

Divide $(x^2 + 5x + 6)$ by $(x + 2)$

Set up long division: Write $x + 2$ outside and $x^2 + 5x + 6$ inside = Division setup ready

Divide first terms: $\frac{x^2}{x} = x$ = First term of quotient: $x$

Multiply and subtract: $x(x + 2) = x^2 + 2x$; subtract from dividend = $(x^2 + 5x) - (x^2 + 2x) = 3x$

Bring down next term: Bring down $+6$ = $3x + 6$

Divide again: $\frac{3x}{x} = 3$ = Second term of quotient: $+3$

Multiply and subtract: $3(x + 2) = 3x + 6$; subtract = $(3x + 6) - (3x + 6) = 0$

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

Synthetic Division

Use synthetic division to divide $(2x^3 - 5x^2 + 3x - 1)$ by $(x - 2)$

Identify c value: For $(x - 2)$, we use $c = 2$ = $c = 2$

Write coefficients: Coefficients: $2, -5, 3, -1$ = Row setup: 2 | 2 -5 3 -1

Bring down first coefficient: Bring down the 2 = First value: 2

Multiply and add: $2 \times 2 = 4$; $-5 + 4 = -1$ = Second value: $-1$

Continue process: $2 \times (-1) = -2$; $3 + (-2) = 1$ = Third value: $1$

Final step: $2 \times 1 = 2$; $-1 + 2 = 1$ = Remainder: $1$

Answer: $2x^2 - x + 1$ with remainder $1$, or $2x^2 - x + 1 + \frac{1}{x-2}$

Mistake: Forgetting to include zero coefficients for missing terms

Why: In $(x^3 - 8)$, the $x^2$ and $x$ terms have coefficient $0$ and must be included in synthetic division.

Correct: Write $(x^3 + 0x^2 + 0x - 8)$ before dividing

Mistake: Using the wrong sign in synthetic division

Why: For $(x - 2)$, use $c = +2$, not $-2$. The sign is opposite to what appears in the divisor.

Correct: $(x - c)$ means use $+c$; $(x + c)$ means use $-c$

Mistake: Incorrect subtraction in long division

Why: Students often add instead of subtract when eliminating terms.

Correct: Always subtract the entire product: change all signs before adding

Engineering: Transfer Functions

Control engineers divide polynomials to analyze system behavior and design stable controllers.

A system's transfer function $\frac{x^2 + 3x + 2}{x + 1}$ simplifies to $x + 2$ after polynomial division.

Computer Graphics: Bezier Curves

Polynomial division helps split curves for rendering and animation in video games and CGI.

Dividing a cubic Bezier curve into smaller segments requires polynomial division at specific parameter values.

Q: When should I use synthetic division vs. long division?

A: Use synthetic division when the divisor is in the form $(x - c)$ with a leading coefficient of 1. For any other divisor (like $2x + 3$ or $x^2 + 1$), use long division.

Q: How do I check if my polynomial division is correct?

A: Multiply: Quotient $\times$ Divisor $+$ Remainder should equal the original Dividend. For example, if $(x^2 + 5x + 6) \div (x + 2) = x + 3$, then $(x + 3)(x + 2) = x^2 + 5x + 6$.

Q: What if the remainder is not zero?

A: Write the answer as: Quotient $+ \frac{\text{Remainder}}{\text{Divisor}}$. For example: $2x + 3 + \frac{5}{x-1}$.