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Page 1
Q u e s t i o n : 5 3
Using factor theorem, factorize each of the following polynomials:
x
3
+ 6x
2
+ 11x + 6
S o l u t i o n :
Let f(x) = x
3
+ 6x
2
+ 11x + 6 be the given polynomial.
Now, put the we get
Therefore, is a factor of f(x).
Now,
f x = x
3
+5x
2
+x
2
+5x +6x +6
Hence, are the factors of f(x).
Q u e s t i o n : 5 4
x
3
+ 2x
2
- x - 2
S o l u t i o n :
Let be the given polynomial.
Now, put the x = -1, we get
f -1 = (-1)
3
+2(-1)
2
-(-1)-2 = 0
Therefore, is a factor of polynomial f(x).
Now, x
3
+ 2x
2
- x - 2 can be written as,
f x = x
3
+3x
2
-x
2
-3x +2x -2
Hence, are the factors of the polynomial f
x.
Q u e s t i o n : 5 5
x
3
- 6x
2
+ 3x + 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
f x = x
3
-7x
2
+x
2
+10x -7x +10
Hence, are the factors of the polynomial f(x).
Q u e s t i o n : 5 6
( )
( )
( )
( )
Page 2
Q u e s t i o n : 5 3
Using factor theorem, factorize each of the following polynomials:
x
3
+ 6x
2
+ 11x + 6
S o l u t i o n :
Let f(x) = x
3
+ 6x
2
+ 11x + 6 be the given polynomial.
Now, put the we get
Therefore, is a factor of f(x).
Now,
f x = x
3
+5x
2
+x
2
+5x +6x +6
Hence, are the factors of f(x).
Q u e s t i o n : 5 4
x
3
+ 2x
2
- x - 2
S o l u t i o n :
Let be the given polynomial.
Now, put the x = -1, we get
f -1 = (-1)
3
+2(-1)
2
-(-1)-2 = 0
Therefore, is a factor of polynomial f(x).
Now, x
3
+ 2x
2
- x - 2 can be written as,
f x = x
3
+3x
2
-x
2
-3x +2x -2
Hence, are the factors of the polynomial f
x.
Q u e s t i o n : 5 5
x
3
- 6x
2
+ 3x + 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
f x = x
3
-7x
2
+x
2
+10x -7x +10
Hence, are the factors of the polynomial f(x).
Q u e s t i o n : 5 6
( )
( )
( )
( )
x
4
- 7x
3
+ 9x
2
+ 7x - 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
f 1 = (1)
4
-7(1)
3
+9(1)
2
+7(1)-10 = 1 -7 +9 +7 -10 = 0
Therefore, is a factor of polynomial f(x).
Now,
f x = x
4
-x
3
-6x
3
+6x
2
+3x
2
-3x +10x -10
Where
Putting we get
Therefore, is a factor of g(x).
Now,
g x = x
3
-7x
2
+x
2
-7x +10x +10
From equation
i and
ii, we get
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 7
3x
3
- x
2
- 3x + 1
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 8
x
3
- 23x
2
+ 142x - 120
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 9
y
3
- 7y + 6
S o l u t i o n :
( )
( )
( )
Page 3
Q u e s t i o n : 5 3
Using factor theorem, factorize each of the following polynomials:
x
3
+ 6x
2
+ 11x + 6
S o l u t i o n :
Let f(x) = x
3
+ 6x
2
+ 11x + 6 be the given polynomial.
Now, put the we get
Therefore, is a factor of f(x).
Now,
f x = x
3
+5x
2
+x
2
+5x +6x +6
Hence, are the factors of f(x).
Q u e s t i o n : 5 4
x
3
+ 2x
2
- x - 2
S o l u t i o n :
Let be the given polynomial.
Now, put the x = -1, we get
f -1 = (-1)
3
+2(-1)
2
-(-1)-2 = 0
Therefore, is a factor of polynomial f(x).
Now, x
3
+ 2x
2
- x - 2 can be written as,
f x = x
3
+3x
2
-x
2
-3x +2x -2
Hence, are the factors of the polynomial f
x.
Q u e s t i o n : 5 5
x
3
- 6x
2
+ 3x + 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
f x = x
3
-7x
2
+x
2
+10x -7x +10
Hence, are the factors of the polynomial f(x).
Q u e s t i o n : 5 6
( )
( )
( )
( )
x
4
- 7x
3
+ 9x
2
+ 7x - 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
f 1 = (1)
4
-7(1)
3
+9(1)
2
+7(1)-10 = 1 -7 +9 +7 -10 = 0
Therefore, is a factor of polynomial f(x).
Now,
f x = x
4
-x
3
-6x
3
+6x
2
+3x
2
-3x +10x -10
Where
Putting we get
Therefore, is a factor of g(x).
Now,
g x = x
3
-7x
2
+x
2
-7x +10x +10
From equation
i and
ii, we get
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 7
3x
3
- x
2
- 3x + 1
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 8
x
3
- 23x
2
+ 142x - 120
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 9
y
3
- 7y + 6
S o l u t i o n :
( )
( )
( )
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f (y).
Q u e s t i o n : 6 0
x
3
- 10x
2
- 53x - 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 1
y
3
- 2y
2
- 29y - 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 2
2y
3
- 5y
2
- 19y + 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 3
x
3
+ 13x
2
+ 32x + 20
S o l u t i o n :
Let be the given polynomial.
Page 4
Q u e s t i o n : 5 3
Using factor theorem, factorize each of the following polynomials:
x
3
+ 6x
2
+ 11x + 6
S o l u t i o n :
Let f(x) = x
3
+ 6x
2
+ 11x + 6 be the given polynomial.
Now, put the we get
Therefore, is a factor of f(x).
Now,
f x = x
3
+5x
2
+x
2
+5x +6x +6
Hence, are the factors of f(x).
Q u e s t i o n : 5 4
x
3
+ 2x
2
- x - 2
S o l u t i o n :
Let be the given polynomial.
Now, put the x = -1, we get
f -1 = (-1)
3
+2(-1)
2
-(-1)-2 = 0
Therefore, is a factor of polynomial f(x).
Now, x
3
+ 2x
2
- x - 2 can be written as,
f x = x
3
+3x
2
-x
2
-3x +2x -2
Hence, are the factors of the polynomial f
x.
Q u e s t i o n : 5 5
x
3
- 6x
2
+ 3x + 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
f x = x
3
-7x
2
+x
2
+10x -7x +10
Hence, are the factors of the polynomial f(x).
Q u e s t i o n : 5 6
( )
( )
( )
( )
x
4
- 7x
3
+ 9x
2
+ 7x - 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
f 1 = (1)
4
-7(1)
3
+9(1)
2
+7(1)-10 = 1 -7 +9 +7 -10 = 0
Therefore, is a factor of polynomial f(x).
Now,
f x = x
4
-x
3
-6x
3
+6x
2
+3x
2
-3x +10x -10
Where
Putting we get
Therefore, is a factor of g(x).
Now,
g x = x
3
-7x
2
+x
2
-7x +10x +10
From equation
i and
ii, we get
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 7
3x
3
- x
2
- 3x + 1
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 8
x
3
- 23x
2
+ 142x - 120
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 9
y
3
- 7y + 6
S o l u t i o n :
( )
( )
( )
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f (y).
Q u e s t i o n : 6 0
x
3
- 10x
2
- 53x - 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 1
y
3
- 2y
2
- 29y - 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 2
2y
3
- 5y
2
- 19y + 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 3
x
3
+ 13x
2
+ 32x + 20
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 4
x
3
- 3x
2
- 9x - 5
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 5
2y
3
+ y
2
- 2y - 1
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 6
x
3
- 2x
2
- x + 2
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 7
Factorize each of the following polynomials:
i
x
3
+ 13x
2
+ 31x - 45 given that x + 9 is a factor
ii
4x
3
+ 20x
2
+ 33x + 18 given that 2x + 3 is a factor.
Page 5
Q u e s t i o n : 5 3
Using factor theorem, factorize each of the following polynomials:
x
3
+ 6x
2
+ 11x + 6
S o l u t i o n :
Let f(x) = x
3
+ 6x
2
+ 11x + 6 be the given polynomial.
Now, put the we get
Therefore, is a factor of f(x).
Now,
f x = x
3
+5x
2
+x
2
+5x +6x +6
Hence, are the factors of f(x).
Q u e s t i o n : 5 4
x
3
+ 2x
2
- x - 2
S o l u t i o n :
Let be the given polynomial.
Now, put the x = -1, we get
f -1 = (-1)
3
+2(-1)
2
-(-1)-2 = 0
Therefore, is a factor of polynomial f(x).
Now, x
3
+ 2x
2
- x - 2 can be written as,
f x = x
3
+3x
2
-x
2
-3x +2x -2
Hence, are the factors of the polynomial f
x.
Q u e s t i o n : 5 5
x
3
- 6x
2
+ 3x + 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
f x = x
3
-7x
2
+x
2
+10x -7x +10
Hence, are the factors of the polynomial f(x).
Q u e s t i o n : 5 6
( )
( )
( )
( )
x
4
- 7x
3
+ 9x
2
+ 7x - 10
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
f 1 = (1)
4
-7(1)
3
+9(1)
2
+7(1)-10 = 1 -7 +9 +7 -10 = 0
Therefore, is a factor of polynomial f(x).
Now,
f x = x
4
-x
3
-6x
3
+6x
2
+3x
2
-3x +10x -10
Where
Putting we get
Therefore, is a factor of g(x).
Now,
g x = x
3
-7x
2
+x
2
-7x +10x +10
From equation
i and
ii, we get
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 7
3x
3
- x
2
- 3x + 1
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 8
x
3
- 23x
2
+ 142x - 120
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 5 9
y
3
- 7y + 6
S o l u t i o n :
( )
( )
( )
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f (y).
Q u e s t i o n : 6 0
x
3
- 10x
2
- 53x - 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 1
y
3
- 2y
2
- 29y - 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 2
2y
3
- 5y
2
- 19y + 42
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(y).
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 3
x
3
+ 13x
2
+ 32x + 20
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 4
x
3
- 3x
2
- 9x - 5
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 5
2y
3
+ y
2
- 2y - 1
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Now,
Hence are the factors of polynomial f(y).
Q u e s t i o n : 6 6
x
3
- 2x
2
- x + 2
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 7
Factorize each of the following polynomials:
i
x
3
+ 13x
2
+ 31x - 45 given that x + 9 is a factor
ii
4x
3
+ 20x
2
+ 33x + 18 given that 2x + 3 is a factor.
S o l u t i o n :
i Let be the given polynomial.
is a factor of the polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
ii Let be the given polynomial.
Therefore is a factor of the polynomial f(x).
Now,
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 8
x
4
- 2x
3
- 7x
2
+ 8x + 12
S o l u t i o n :
Let be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
f x = x
4
-3x
3
+x
3
-3x
2
-4x
2
+12x -4x +12
Where
Putting we get
Therefore, is the factor of g(x).
Now,
g x = x
3
-2x
2
-x
2
-6x +2x +12
From equation
i and
ii, we get
Hence are the factors of polynomial f(x).
Q u e s t i o n : 6 9
x
4
+ 10x
3
+ 35x
2
+ 50x + 24
S o l u t i o n :
Let
f(x) = x
4
+10x
3
+35x
2
+50x +24 be the given polynomial.
Now, putting we get
Therefore, is a factor of polynomial f(x).
Now,
( )
( )
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Students of Class 9 can study Factorization of Polynomials - Exercise 6.5 alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Factorization of Polynomials - Exercise 6.5,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Factorization of Polynomials - Exercise 6.5 is prepared as per the latest Class 9 syllabus.
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