Notes on Division of Polynomials

Polynomials and Polynomial Functions

A polynomial is a mathematical expression built from variables and constants using only the operations of addition, subtraction, and multiplication, with nonnegative integer exponents on the variables.
A polynomial expresses a relationship between variables and their coefficients.
A polynomial in one variable x is typically written as a finite sum of terms of the form ai x^i, where i is a nonnegative integer and ai are constants (real numbers).

A polynomial function is a function defined by a polynomial in one variable: P(x)=an x^n+a{n-1} x^{n-1}+\cdots+a1 x+a0, where a_n\neq 0, and the degree is n (the highest exponent with a nonzero coefficient).
If there are real numbers denoted by a, the function with one variable and degree n can be written in the standard form above.

The degree of a polynomial is the highest exponent of the variable in the polynomial.
Example: If P(x)=an x^n+…+a1 x+a0, then \deg(P)=n provided an\neq 0.
The leading term is the term with the highest power of the variable: a_n x^n.
The degree gives information about behavior for large |x| and helps classify polynomials.

Expressions with nonpolynomial features (e.g., square roots of variables, variable in the denominator, fractional exponents, trigonometric functions of a variable) are generally not polynomials.
Examples of not-polynomials include expressions with roots of x, x^(-1), or sin(x).

Goal: Express a dividend P(x) as P(x) = D(x) · Q(x) + R(x) where D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder.
The Division Algorithm for Polynomials: P(x)=D(x)\cdot Q(x)+R(x)\quad\text{with}\quad \deg(R)<\deg(D).
If the divisor is constant (numerical), this reduces to ordinary integer division rules.

Setup: Write dividend in descending powers of the variable and place divisor beside it.
Step 1: Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
Step 2: Multiply the entire divisor by that term and subtract the result from the dividend.
Step 3: Bring down the next term (if needed) and repeat Step 1–2 until the degree of the remainder is less than the degree of the divisor.
Step 4: The remaining expression is the remainder R(x). The collected terms form the quotient Q(x).
Key reminder: Always arrange all terms in decreasing powers of the variable; fill any missing degrees with zero.

Divide: P(x)=x^3+2x-7 by D(x)=x-2.
Using long division or synthetic, you obtain:
- Quotient: Q(x)=x^2+2x+6
- Remainder: R=5
Relationship: P(x)=(x-2)\cdot(x^2+2x+6)+5.

Synthetic division is a shortcut method when dividing by a binomial of the form x-c (i.e., divisor monic with zero constant term after factoring out any common factors).
Steps (for divisor x - c):
1) Write the coefficients of the dividend in descending powers of x.
2) Bring down the leading coefficient as the first coefficient of the quotient.
3) Multiply c by the value just written on the bottom row and write it under the next coefficient.
4) Add the two numbers in that column and write the sum on the bottom row.
5) Repeat steps 3–4 until all coefficients are processed.
6) The bottom row (except the final value) gives the quotient coefficients; the final value is the remainder.
Synthetic division is often faster for divisors of the form x - c.

Divide: 2x^3+0x^2+2x-7 by x-2. (Equivalently, divisor c = 2, root x=2.)
- Coefficients: 1, 0, 2, -7
- Synthetic row with c = 2: bring down 1 → multiply 2·1 = 2; add to 0 → 2; multiply 2·2 = 4; add to 2 → 6; multiply 2·6 = 12; add to -7 → 5.
- Quotient coefficients: 1, 2, 6 → x^2+2x+6. Remainder: 5.
Result: P(x)=(x-2)(x^2+2x+6)+5.