Remainder Theorem

Definition: If a polynomial p(x) is divided by a linear polynomial (x − a), then the remainder is p(a).

To find the remainder, substitute x = a into p(x). The resulting value is the remainder.

Mathematical form:

p(x) = (x − a) × q(x) + r

Where:

• q(x) = quotient polynomial
• r = remainder (a constant, since the divisor has degree 1)

Proof:

1. Put x = a in the equation: p(a) = (a − a) × q(a) + r
2. p(a) = 0 × q(a) + r
3. p(a) = r

Extensions:

• Dividing by (x + a): remainder = p(−a), since x + a = x − (−a)
• Dividing by (ax − b): remainder = p(b/a), since the zero of (ax − b) is x = b/a
• Dividing by (ax + b): remainder = p(−b/a)

Important: The Remainder Theorem gives only the remainder, not the quotient. For the quotient, use long division or synthetic division.

Proof of the Remainder Theorem:

Step 1: State the Division Algorithm

• When p(x) is divided by d(x), we get quotient q(x) and remainder r(x) such that:
• p(x) = d(x) × q(x) + r(x)
• where degree of r(x) < degree of d(x), or r(x) = 0.

Step 2: Apply to division by (x − a)

• Let d(x) = (x − a), which has degree 1.
• The remainder r(x) must have degree less than 1, so r(x) is a constant r.
• p(x) = (x − a) × q(x) + r ... (equation 1)

Step 3: Substitute x = a

1. p(a) = (a − a) × q(a) + r
2. p(a) = 0 × q(a) + r
3. p(a) = r

Step 4: Conclusion

• The remainder r = p(a). This proves the Remainder Theorem.

Geometric understanding: When x = a, the factor (x − a) becomes zero, eliminating the quotient term. What remains is the constant remainder.

Extension to (ax − b):

1. The zero of (ax − b) is x = b/a.
2. p(x) = (ax − b) × q(x) + r
3. Put x = b/a: p(b/a) = (a × b/a − b) × q(b/a) + r = 0 + r
4. So remainder = p(b/a).

Types of Remainder Theorem Problems:

• Type 1: Finding the remainder directly — Given p(x) and divisor (x − a), compute p(a). No division needed.
• Type 2: Division by (x + a) — Rewrite as (x − (−a)) and find remainder as p(−a). Remember to use the negative value.
• Type 3: Division by (ax − b) or (ax + b) — Find the zero of the divisor. For (ax − b), zero = b/a. For (ax + b), zero = −b/a.
• Type 4: Finding unknown coefficients — If the remainder is given, set up the equation p(a) = remainder and solve for the unknown.
• Type 5: Checking if a linear expression is a factor — If p(a) = 0, the divisor is a factor. This connects to the Factor Theorem.
• Type 6: Finding remainders for multiple divisors — Evaluate the polynomial at different values for different divisors.
• Type 7: Verification problems — Verify a remainder by both the Remainder Theorem and polynomial long division.

Applications of the Remainder Theorem:

• Quick polynomial evaluation: Provides a fast, structured way to evaluate polynomials at specific values.
• Factorisation: By testing values and finding which give remainder zero, we identify factors of the polynomial (trial-and-error method).
• Checking divisibility: To check if a linear expression divides a polynomial exactly, evaluate the polynomial at the zero of the divisor.
• Finding unknown coefficients: Given conditions about remainders or factors, set up equations using p(a) = R and solve for unknowns.
• Synthetic division (Horner's method): The Remainder Theorem underlies this efficient algorithm for polynomial evaluation used in computer science.
• Error-correcting codes: Polynomial remainder calculations are fundamental to cyclic redundancy check (CRC) codes in data transmission.
• Interpolation: Related to polynomial interpolation methods used in data science and engineering.