Mathematics 11 (Punjab Board)
Chapter 9: Division of Polynomials — Theory
Just as numbers can be divided with a quotient and a remainder, so can polynomials. This chapter builds the tools — long division, the remainder theorem, the factor theorem, and synthetic division — and then uses them on real-life problems.
Introduction: The Division Algorithm
When a polynomial \(f(x)\) (the dividend) is divided by a non-zero polynomial \(g(x)\) (the divisor), there exist unique polynomials \(q(x)\) (the quotient) and \(r(x)\) (the remainder) such that:
Matlab: remainder ki degree hamesha divisor ki degree se chhoti hoti hai. Agar divisor \(x - a\) jaisa linear hai, to remainder sirf aik constant (degree 0) hoga.
Key point: Always check your division by multiplying back: divisor × quotient + remainder must equal the dividend.
9.1 Polynomial Function
A polynomial function has the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where the \(a_i\) are constants and \(n\) is a non-negative integer called the degree. A number \(c\) with \(f(c) = 0\) is called a zero (or root) of the polynomial.
Polynomials are divided by long division: divide the leading terms, multiply back, subtract, bring down the next term, and repeat — exactly like numerical long division.
9.1.1 Remainder Theorem
The remainder theorem finds the remainder without doing the division:
Proof idea: write \(f(x) = (x-a)q(x) + r\) where \(r\) is a constant; putting \(x = a\) kills the first term, leaving \(f(a) = r\).
For a divisor like \(x + 3\), write it as \(x - (-3)\) and evaluate \(f(-3)\).
Key point: Remainder theorem sirf \((x - a)\) jaisay linear divisors par lagta hai — sign ka khayal rakhen: \(x + 3\) ka matlab \(a = -3\).
9.1.2 Factor Theorem
The factor theorem is the remainder theorem with remainder zero:
Is se do kaam hote hain: (1) check karna ke koi expression factor hai ya nahi — bas value put karke dekhen; (2) unknown coefficients nikalna — factor hone ki shart se equations banti hain.
Key point: "Is \(x+1\) a factor of \(f(x)\)?" ka jawab sirf \(f(-1)\) nikal kar mil jata hai — poori division ki zaroorat nahi.
9.1.3 Synthetic Division
Synthetic division is a shortcut for dividing by \(x - a\): write down the coefficients, bring down the first, multiply by \(a\), add to the next coefficient, and repeat. The last number in the bottom row is the remainder; the rest are the quotient's coefficients.
Example: dividing \(x^3 - 7x + 6\) by \(x - 2\) (coefficients \(1, 0, -7, 6\)):
Remainder 0, quotient \(x^2 + 2x - 3\). Missing powers ke liye coefficient 0 likhna na bhoolen (yahan \(x^2\) ka coefficient 0 tha).
Key point: Synthetic division ke baad quotient ko mazeed factorize karke polynomial ko completely factor kiya ja sakta hai.
9.2 Real-Life Applications of the Remainder and Factor Theorems
Polynomial models appear in data fitting (polynomial regression), where \(P(x)\) predicts a value and the percentage error against an observed value is:
In signal processing, a transfer function like \(H(z) = \frac{N(z)}{D(z)}\) has zeros where \(N(z) = 0\) and poles where \(D(z) = 0\), both found with the factor theorem. A discrete system is stable when every pole lies inside the unit circle, i.e. \(|z| \lt 1\) for each pole.
Key point: Zeros numerator se, poles denominator se — aur stability ka test sirf poles ki absolute values ka hai.