Chapter 2: Polynomials

Remainder Theorem

The Powerful Shortcut!

Remainder Theorem: When p(x)p(x) is divided by (x−a)(x - a), the remainder is simply p(a)p(a).

No long division needed - just substitute!

Why This Works

When we divide p(x)p(x) by (x−a)(x - a):

p(x)=(x−a)×q(x)+rp(x) = (x - a) \times q(x) + r

Where:

  • q(x)q(x) = quotient
  • rr = remainder (a constant)

Putting x=ax = a: p(a)=(a−a)×q(a)+rp(a) = (a - a) \times q(a) + r p(a)=0+rp(a) = 0 + r p(a)=rp(a) = r ✓

Example 1

Find remainder when p(x)=x3+3x2−5x+7p(x) = x^3 + 3x^2 - 5x + 7 is divided by (x−2)(x - 2).

Step 1: Identify aa From (x−2)(x - 2), we have a=2a = 2

Step 2: Calculate p(2)p(2) p(2)=(2)3+3(2)2−5(2)+7p(2) = (2)^3 + 3(2)^2 - 5(2) + 7 =8+12−10+7= 8 + 12 - 10 + 7 =17= 17

Remainder = 17 ✓

Example 2

Find remainder when divided by (x+3)(x + 3).

Key insight: (x+3)=(x−(−3))(x + 3) = (x - (-3)), so a=−3a = -3

Example 3

Find remainder when divided by (2x−1)(2x - 1).

Key insight: Set 2x−1=0⇒x=122x - 1 = 0 \Rightarrow x = \frac{1}{2}

So substitute x=12x = \frac{1}{2} into p(x)p(x)!

Visualizer
Step-by-Step Solution
1
p(x) = x³ + 3x² - 5x + 7
Given polynomial
2
Divisor: (x - 2), so a = 2
Find the value to substitute
3
p(2) = (2)³ + 3(2)² - 5(2) + 7
Substitute x = 2
4
= 8 + 12 - 10 + 7
Calculate each term
5
= 17
Remainder is 17!

Interactive Visualization