Factorization • Topic 2 of 3

Remainder Theorem

What is the Remainder Theorem? The Remainder Theorem tells us the remainder when a polynomial is divided by a linear factor without actually performing the division. It states:

When a polynomial p(x) is divided by (x − a), the remainder is p(a).

General form: If p(x) is divided by (x − a), then: p(x) = (x − a) × q(x) + r, where r = p(a) (the remainder)

Key differences from Factor Theorem: | | Remainder Theorem | Factor Theorem | |---|---|---| | Gives | Remainder when dividing by (x−a) | Whether (x−a) is a factor | | Condition | Always works | Works only if remainder = 0 | | Formula | Remainder = p(a) | p(a) = 0 ⇔ (x−a) is factor |

Real-life analogy: Imagine sharing cookies among friends. If you have p(x) cookies and (x−a) friends, the remainder theorem tells you how many cookies will be left over without actually sharing them!

Important extensions:

  • Divisor (ax − b): Remainder = p(b/a)
  • Divisor (ax + b): Remainder = p(−b/a)
Division identity p of x equals x minus a times q of x plus r with remainder r equal to p of a Remainder Theorem: r = p(a) p(x) = (x − a) · q(x) + r dividend = divisor · quotient + remainder divisor (x − a) remainder r put x = a to kill this term r = p(a) If r = 0 then (x − a) is a factor (Factor Theorem) Remainder when 2x cubed minus 3x squared plus 5x minus 7 is divided by x minus 3 Worked example: divide by (x − 3) p(x) = 2x^3 − 3x^2 + 5x − 7,   a = 3 r = p(3) = 2(27) − 3(9) + 5(3) − 7 = 54 − 27 + 15 − 7 remainder = 35
Solved Examples
1
Worked Example
Find the remainder when p(x) = \(x^{3} - 2x^{2} + 3x - 4\) is divided by (x − 2).
Solution
  1. Step 1: Using Remainder Theorem, remainder = p(2)
  2. Step 2: p(2) = (2)\(^{3} - 2\)(2)\(^{2} + 3\)(2) − 4 = 8 − 8 + 6 − 4
  3. Step 3: 8 − 8 = 0, 0 + 6 = 6, 6 − 4 = 2

Answer: Remainder = 2

2
Worked Example
Find the remainder when p(x) = \(2x^{4} - 5x^{3} + x^{2} - 3x + 7\) is divided by (2x + 1).
Solution
  1. Step 1: For divisor (2x + 1), set 2x + 1 = 0 → x = −1/2
  2. Step 2: Remainder = p(−1/2)
  3. Step 3: p(−1/2) = 2(1/16) − 5(−1/8) + (1/4) − 3(−1/2) + 7
  4. Step 4: = 2/16 + 5/8 + 1/4 + 3/2 + 7
  5. Step 5: Convert to denominator 16: 2/16 + 10/16 + 4/16 + 24/16 + 112/16
  6. Step 6: = (2+10+4+24+112)/16 = 152/16 = 19/2 = 9.5

Answer: Remainder = 19/2 (or 9.5)

3
Worked Example
The polynomial p(x) = \(x^{3} + 2x^{2} - 5x + k\) leaves remainder 10 when divided by (x − 2). Find k.
Solution
  1. Step 1: Using Remainder Theorem, remainder = p(2)
  2. Step 2: p(2) = 8 + 8 − 10 + k = 6 + k
  3. Step 3: Given remainder = 10, so 6 + k = 10
  4. Step 4: k = 10 − 6 = 4

Answer: k = 4

Key Points

  • Remainder Theorem: When p(x) is divided by (x − a), remainder = p(a)
  • No division needed — just substitute the value
  • For divisor (ax − b), remainder = p(b/a)
  • For divisor (ax + b), remainder = p(−b/a)
  • Factor Theorem is a special case of Remainder Theorem (when remainder = 0)
  • Remainder can be zero, positive, negative, or fractional
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The remainder when $p(x)$ is divided by $(x-a)$ is:
Explanation: Remainder theorem.
Q2.The remainder of $p(x)\div(x+a)$ is:
Explanation: $x+a=0\Rightarrow x=-a$.
Q3.If the remainder is $0$, then $(x-a)$ is a:
Explanation: A factor.
Q4.The remainder of $x^2$ divided by $(x-1)$ is:
Explanation: $p(1)=1$.
Practice more MCQs → 📝 Assignment · 35 marks