Factor Theoem Factorization Notes Class 10 OP Malhotra Maths (2026-27). We Provide Step by Step Answer of all the exercises with Chapter Test of S Chand OP Malhotra Maths . Visit official Website CISCE  for detail information about ICSE Board Class-10.

Factor Theoem Factorization Notes Class 10 OP Malhotra Maths (2026-27)

Rational Integral Function

A polynomial in x , in which no term contains a fractional power of x , or a negative power of x , is called a rational integral function of x , and is denoted by f(x) or F(x) or Q(x) etc.

For example , 3x⁴– 5x³+2x+1 , 2x³+7x²-x+4 are rational integral function of x . They can be written as
f(x) = 3x⁴– 5x³+2x+1 , F(x) = 2x³+7x²-x+4

Value of a Function

The value of a function f(x) or F(x) for x=a is denoted by f(a) or F(a) , as the case may be , and it is obtained by putting x=a in the polynomial.

If f(x) = 3x⁴– 5x³+2x+1 , then f(2) = (3×2⁴)-(5×2³)+(2×2)+1 = 48-40+4+1 = 13
If F(x) = 2x³+7x²-x+4 , then f(-1) = (2×(-1)³)+7(-1)²-(-1)+4 = -2+7+1+4 = 10

Factor Theorm

If p(x) is a polynomial and p(a)=0 , then (x-a) is a factor of p(x).

Conversely (x-a) is a factor p(x) , then p(a)=0 .

Alternative Form of the factor theorem 

Case 1: If f(-a) = 0 , then (x+a) is a factor f(x)

Case 2: If f(-a/b) = 0 , then (bx+a) is a factor f(x)

Case 3:  If p(a/b)= 0, then (bx-a) is a factor of f(x).

Case 4: If f(a)=0 and f(b) =0, then (x-a) and (x-b) are factors of f(x).

Remainder Theorem

Let f(x) be any rational integral function of x or polynomial of degree greater than or equal to 1 (≥1) and ‘a’ be any real number. If f(x) is divided (x-a), then the remainder is always equal to f(a).

Case 1: If f(x) is divided by (x+a) , the remainder is f(-a).

Case 2: If f(x) is divided by ax-b , the remainder is f(b/a).

Case 3: If f(x) is divided by ax+b , the remainder is f(-b/a).

Important Points to Remember 

• p(a) = 0 ⇒ (x-a) is a factor of p(x)
• p(a) ≠ 0 ⇒ (x-a) is not a  factor of p(x)
• Factor Theorem is used mainly to factorize cubic and higher-degree polynomials.
• Remainder Theorem helps in quickly checking factors.

Some worked out examples

Q1. Find the remainder when f(x) = x³ − 4x + 5 is divided by:

1. x − 2
2. x + 1
3. 2x − 1

Sol: By the Remainder Theorem, when f(x) is divided by (x − a), the remainder is f(a).

(i) Divisor: x − 2
Remainder = f(2)
= 2³ − 4(2) + 5
= 8 − 8 + 5
= 5

Required remainder = 5

(ii) Divisor: x + 1
Remainder = f(−1)
= (−1)³ − 4(−1) + 5
= −1 + 4 + 5
= 8

Required remainder = 8

(iii) Divisor: 2x − 1
2x − 1 = 0
x = 1/2

Remainder = f(1/2)
= (1/2)³ − 4(1/2) + 5
= 1/8 − 2 + 5
= 25/8

Required remainder = 25/8

Q2. Find the remainder when f(x) = 2x³ − 5x² + x + 3 is divided by (1 − 2x).

Sol: By the Remainder Theorem,
1 − 2x = 0
x = 1/2

Therefore,
Remainder = f(1/2)
= 2(1/2)³ − 5(1/2)² + 1/2 + 3
= 1/4 − 5/4 + 1/2 + 3
= −1 + 1/2 + 3
= 5/2

Hence, the required remainder is 5/2.

Q3. If the polynomial f(x) = 3x³ − ax² + 2x − 5 leaves a remainder −45 when divided by (x + 1), find the value of a.

Sol: By the Remainder Theorem,
f(−1) = −45
3(−1)³ − a(−1)² + 2(−1) − 5 = −45
−3 − a − 2 − 5 = −45
−10 − a = −45

a = 35

Hence, a = 35.

Q4, Use Factor Theorem to determine whether x + 2 is a factor of f(x) = x³ − 4x² + 4x + 16

Sol: By Factor Theorem,
x + 2 is a factor if f(−2) = 0.

f(−2) = (−2)³ − 4(−2)² + 4(−2) + 16
= −8 − 16 − 8 + 16
= −16

Since f(−2) ≠ 0,

x + 2 is not a factor.

Q5. Find the value of a if x + 2 is a factor of f(x) = x³ + ax² − 3x + 2

Sol: Since x + 2 is a factor,
f(−2) = 0
(−2)³ + a(−2)² − 3(−2) + 2 = 0
⇒ −8 + 4a + 6 + 2 = 0
⇒ 4a = 0
⇒ a = 0

Hence, a = 0.

Q6. Factorize x³ + 8x² + 7x − 90 given that x + 9 is a factor.

Sol: Dividing by (x + 9), we get\
x³ + 8x² + 7x − 90 = (x + 9)(x² − x − 10)

Now,
x² − x − 10 = (x − 5)(x + 2)

Therefore,
x³ + 8x² + 7x − 90 = (x + 9)(x − 5)(x + 2)

Hence, the required factorization is (x + 9)(x − 5)(x + 2).

Practice Questions on Factor Theorem – Factorization : Exercise-7

In this chapter, we study all the topics on Factor Theorem and do some practice questions also. Here we solve extra practice questions on this chapter for better understanding.

Here is the link for extra practice questions on Factor Theorem :- Self Evaluation

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