Ex

  Mathematics (NTSE/Olympiad)  

Polynomial

Remainder Theorem

Important Points of a Remainder Theorem

(i) Remainder can be obtained on dividing polynomial p(x) by x – a is equal to p(a).
(ii) If a polynomial p(x) is divided by (x + a) the remainder is the value of p(x) at x = –a.
(iii) (x – a) is a factor of polynomial p(x) if p(a) = 0
(iv) (x + a) is a factor of polynomial p(x) if p(–a) = 0
(v) (x – a) (x – b) is a factor of polynomial p(x), if p(a) = 0 and p(b) = 0.

Example: Find the remainder when 4x3 – 3x2 + 2x – 4 is divided by
(a) x – 1 (b) x + 2 (c) x + 1/2  

Solution:

Let the given cubic equation is p(x) = 4x3 – 3x2 + 2x – 4
(a) When p(x) is divided by (x – 1), then by remainder theorem, the required remainder will be p(1)
p(1) = 4 (1)3 – 3(1)2 + 2(1) – 4
= 4 × 1 – 3 × 1 + 2 × 1 – 4
= 4 – 3 + 2 – 4 = – 1

(b) When p(x) is divided by (x + 2), then by remainder theorem, the required remainder will be p (–2).
p(–2) = 4 (–2)3 – 3 (–2)2 + 2(–2) – 4
= 4 × (–8) – 3 × 4 – 4 – 4
= – 32 – 12 – 8 = – 52

(c) When p(x) is divided by x + 1/2, then by remainder theorem, the required remainder will be
p(–1/2)= 4 (–1/2)3 – 3(–1/2)2 + 2(–1/2) – 4
= 4 × (–1/8) – 3 (1/4)× – 2 × (1/2) – 4
= –1/2 – 3/4 – 1– 4 = –1/2 –3/4  – 5
= (–2 – 3 – 20)/4 = –25/4


If you want to give information about online courses to other students, then share it with more and more on Facebook, Twitter, Google Plus. The more the shares will be, the more students will benefit. The share buttons are given below for your convenience.
×

NTSE Mathematics (Class X)

  • Trigonometry
  • Similar Triangles
  • Statistics
  • Quadratic Equation
  • Arithmetic Progressions
  • Application of Trigonometry
  • Circle
  • Co-ordinate Geometry
  • Area related to Circle
  • Surface Area & Volume
  • Constructions
  • Probability

NTSE Mathematics (Class IX)

  • Trigonometry
  • Similar Triangles
  • Statistics
  • Quadratic Equation
  • Arithmetic Progressions
  • Application of Trigonometry
  • Circle
  • Co-ordinate Geometry
  • Area related to Circle
  • Surface Area & Volume
  • Constructions
  • Probability

SHOW CHAPTERS

NTSE Physics Course (Class 9 & 10)

NTSE Chemistry Course (Class 9 & 10)

NTSE Geography Course (Class 9 & 10)

NTSE Biology Course (Class 9 & 10)

NTSE Democratic Politics Course (Class 9 & 10)

NTSE Economics Course (Class 9 & 10)

NTSE History Course (Class 9 & 10)

NTSE Mathematics Course (Class 9 & 10)