Polynomial Long Division
Dividing 137 by 4 asks "how many 4's fit into 137?" — answer 34, remainder 1. Polynomial long division is the same question with variables: how many times does the divisor fit into the dividend? You get a quotient polynomial plus a remainder whose degree is smaller than the divisor's. The only change from arithmetic is that you compare the highest power of the variable instead of place value.
For polynomials P(x) and D(x)=0 there are unique Q(x) and R(x) with
P(x)=D(x)Q(x)+R(x),degR<degD.
This is the Division Algorithm for Polynomials.
The routine
To divide P(x) by D(x), repeat until the remainder's degree drops below degD:
- Divide the leading term of the current dividend by the leading term of D(x) — this is the next quotient term.
- Multiply the whole divisor by that term.
- Subtract to get a new, lower-degree dividend, then repeat.
For example, dividing 2x3+3x2−5x+1 by x−2: the successive quotient terms are 2x2, then 7x, then 9, leaving remainder 19. So
2x3+3x2−5x+1=(x−2)(2x2+7x+9)+19.
The remainder 19 has degree 0<1, exactly as the algorithm requires.
Insert zero coefficients for missing terms — write x3+1 as x3+0x2+0x+1 — or the columns misalign during subtraction.
Why it matters
- If R(x)=0, then D(x) is a factor of P(x).
- Dividing by (x−a) leaves remainder P(a) — the Remainder Theorem (here P(2)=19).
- It reduces an improper rational function to a polynomial plus a proper fraction — the first step before partial fractions or integration.
Polynomial long division is introduced as early as the NCERT Class 9-10 Polynomials chapters and resurfaces as an essential prerequisite skill in the Class 12 Integrals chapter, wherever an improper rational function needs to be simplified before integration or partial fractions. Students searching 'polynomial long division examples class 10' or 'division algorithm for polynomials' will find this quotient-and-remainder method is exactly the same one tested in board exams at both levels.