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Factorization of Polynomials - Exercise 6.5 | Extra Documents & Tests for Class 9 PDF Download

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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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