Class 9 Exam > Class 9 Notes > Mathematics (Maths) Class 9 > RD Sharma Solutions: Factorization of Polynomials- 2
Factorization of Polynomials- 2 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download
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Page 1
Q u e s t i o n : 9
If fx
= 2x
3
- 13x
2
+ 17x + 12, find i
f2
ii
f-3
iii
f0
S o l u t i o n :
Let be the given polynomial
i The value of f
2 can be found by putting x = 2
= 10
ii The value of f
– 3 can be found by putting x = –3
iii The value of f
0 can be found by putting x = 0
Q u e s t i o n : 1 0
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:
i
( )
Page 2
Q u e s t i o n : 9
If fx
= 2x
3
- 13x
2
+ 17x + 12, find i
f2
ii
f-3
iii
f0
S o l u t i o n :
Let be the given polynomial
i The value of f
2 can be found by putting x = 2
= 10
ii The value of f
– 3 can be found by putting x = –3
iii The value of f
0 can be found by putting x = 0
Q u e s t i o n : 1 0
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:
i
( )
f x = 3x +1, x = -
1
3
ii
f x = x
2
-1, x = 1, -1
iii
g x = 3x
2
-2, x =
2
v
3
, -
2
v
3
iv
p x = x
3
-6x
2
+11x -6, x = 1, 2, 3
v
f x = 5x - p, x =
4
5
vi
f x = x
2
, x = 0
vii
f x = lx +m, x = -
m
1
viii
f x = 2x +1, x =
1
2
S o l u t i o n :
i To check whether the given number is the zero of the polynomial or not we have to find
Hence, is the zeros of the given polynomial.
ii To check whether the given number is the zero of the polynomial or not we have to find and
Now,
And ,
Hence, and x = 1 are the zeros of the polynomial.
iii To check whether the given number is the zero of the polynomial or not we have to find
Now,
And
Here, both the value of , are not satisfied the polynomial. Therefore, they are not the zeros of the polynomial.
iv To check whether the given number is the zero of the polynomial or not we have to find
( )
( )
( )
( )
( )
( )
( )
( )
Page 3
Q u e s t i o n : 9
If fx
= 2x
3
- 13x
2
+ 17x + 12, find i
f2
ii
f-3
iii
f0
S o l u t i o n :
Let be the given polynomial
i The value of f
2 can be found by putting x = 2
= 10
ii The value of f
– 3 can be found by putting x = –3
iii The value of f
0 can be found by putting x = 0
Q u e s t i o n : 1 0
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:
i
( )
f x = 3x +1, x = -
1
3
ii
f x = x
2
-1, x = 1, -1
iii
g x = 3x
2
-2, x =
2
v
3
, -
2
v
3
iv
p x = x
3
-6x
2
+11x -6, x = 1, 2, 3
v
f x = 5x - p, x =
4
5
vi
f x = x
2
, x = 0
vii
f x = lx +m, x = -
m
1
viii
f x = 2x +1, x =
1
2
S o l u t i o n :
i To check whether the given number is the zero of the polynomial or not we have to find
Hence, is the zeros of the given polynomial.
ii To check whether the given number is the zero of the polynomial or not we have to find and
Now,
And ,
Hence, and x = 1 are the zeros of the polynomial.
iii To check whether the given number is the zero of the polynomial or not we have to find
Now,
And
Here, both the value of , are not satisfied the polynomial. Therefore, they are not the zeros of the polynomial.
iv To check whether the given number is the zero of the polynomial or not we have to find
( )
( )
( )
( )
( )
( )
( )
( )
And
Hence, x = 1, 2, 3 are the zeros of the polynomial p(x).
vTo check whether the given number is the zero of the polynomial or not we have to find
Here, , does not satisfy therefore, is not a zero of the polynomial.
viTo check whether the given number is the zero of the polynomial or not we have to find
Hence, x = 0 is the zeros of the polynomial.
vii To check whether the given number is the zero of the polynomial or not we have to find
Hence, is zeros of the polynomial.
viiiTo check whether the given number is the zero of the polynomial or not we have to find
Hence, does not satisyf the polynomial, so, is not zero of the polynomial.
Q u e s t i o n : 1 1
If x = 2 is a root of the polynomial f(x) = 2x
2
- 3x + 7a, find the value of a.
S o l u t i o n :
The given polynomial is
If x = 2 is the root of the polynomial.
Then
Q u e s t i o n : 1 2
If x = -
1
2
is a zero of the polynomial p(x) = 8x
3
- ax
2
-x + 2, find the value of a.
S o l u t i o n :
The given polynomial is
Page 4
Q u e s t i o n : 9
If fx
= 2x
3
- 13x
2
+ 17x + 12, find i
f2
ii
f-3
iii
f0
S o l u t i o n :
Let be the given polynomial
i The value of f
2 can be found by putting x = 2
= 10
ii The value of f
– 3 can be found by putting x = –3
iii The value of f
0 can be found by putting x = 0
Q u e s t i o n : 1 0
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:
i
( )
f x = 3x +1, x = -
1
3
ii
f x = x
2
-1, x = 1, -1
iii
g x = 3x
2
-2, x =
2
v
3
, -
2
v
3
iv
p x = x
3
-6x
2
+11x -6, x = 1, 2, 3
v
f x = 5x - p, x =
4
5
vi
f x = x
2
, x = 0
vii
f x = lx +m, x = -
m
1
viii
f x = 2x +1, x =
1
2
S o l u t i o n :
i To check whether the given number is the zero of the polynomial or not we have to find
Hence, is the zeros of the given polynomial.
ii To check whether the given number is the zero of the polynomial or not we have to find and
Now,
And ,
Hence, and x = 1 are the zeros of the polynomial.
iii To check whether the given number is the zero of the polynomial or not we have to find
Now,
And
Here, both the value of , are not satisfied the polynomial. Therefore, they are not the zeros of the polynomial.
iv To check whether the given number is the zero of the polynomial or not we have to find
( )
( )
( )
( )
( )
( )
( )
( )
And
Hence, x = 1, 2, 3 are the zeros of the polynomial p(x).
vTo check whether the given number is the zero of the polynomial or not we have to find
Here, , does not satisfy therefore, is not a zero of the polynomial.
viTo check whether the given number is the zero of the polynomial or not we have to find
Hence, x = 0 is the zeros of the polynomial.
vii To check whether the given number is the zero of the polynomial or not we have to find
Hence, is zeros of the polynomial.
viiiTo check whether the given number is the zero of the polynomial or not we have to find
Hence, does not satisyf the polynomial, so, is not zero of the polynomial.
Q u e s t i o n : 1 1
If x = 2 is a root of the polynomial f(x) = 2x
2
- 3x + 7a, find the value of a.
S o l u t i o n :
The given polynomial is
If x = 2 is the root of the polynomial.
Then
Q u e s t i o n : 1 2
If x = -
1
2
is a zero of the polynomial p(x) = 8x
3
- ax
2
-x + 2, find the value of a.
S o l u t i o n :
The given polynomial is
If is a zeros of the polynomial p(x).
then
Therefore,
Hence the value of
Q u e s t i o n : 1 3
If x = 0 and x = -1 are the roots of the polynomial fx
=2x
3
- 3x
2
+ ax + b, find the value of a and b.
S o l u t i o n :
The given polynomial is
fx
=2x
3
- 3x
2
+ ax + b
If is zeros of the polynomial f(x), then f
0 = 0
Similarly, if x = - 1 is the zeros of the polynomial of ,
Then,
Putting the value of b from equation
1
Thus,
Q u e s t i o n : 1 4
Find the integral roots of the polynomial fx
= x
3
+ 6x
2
+ 11x + 6.
S o l u t i o n :
The given polynomial is
Here, f(x) is a polynomial with integer coefficient and the coefficient of highest degree term is 1. So, the integer roots of f(x) are factors of 6. Which are by observing.
= 0
Also,
And similarly,
f
-3 = 0
Therefore, the integer roots of the polynomial f(x) are -1, -2, - 3
Q u e s t i o n : 1 5
Find rational roots of the polynomial fx
= 2x
3
+ x
2
- 7x - 6.
S o l u t i o n :
The given polynomial is
Page 5
Q u e s t i o n : 9
If fx
= 2x
3
- 13x
2
+ 17x + 12, find i
f2
ii
f-3
iii
f0
S o l u t i o n :
Let be the given polynomial
i The value of f
2 can be found by putting x = 2
= 10
ii The value of f
– 3 can be found by putting x = –3
iii The value of f
0 can be found by putting x = 0
Q u e s t i o n : 1 0
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:
i
( )
f x = 3x +1, x = -
1
3
ii
f x = x
2
-1, x = 1, -1
iii
g x = 3x
2
-2, x =
2
v
3
, -
2
v
3
iv
p x = x
3
-6x
2
+11x -6, x = 1, 2, 3
v
f x = 5x - p, x =
4
5
vi
f x = x
2
, x = 0
vii
f x = lx +m, x = -
m
1
viii
f x = 2x +1, x =
1
2
S o l u t i o n :
i To check whether the given number is the zero of the polynomial or not we have to find
Hence, is the zeros of the given polynomial.
ii To check whether the given number is the zero of the polynomial or not we have to find and
Now,
And ,
Hence, and x = 1 are the zeros of the polynomial.
iii To check whether the given number is the zero of the polynomial or not we have to find
Now,
And
Here, both the value of , are not satisfied the polynomial. Therefore, they are not the zeros of the polynomial.
iv To check whether the given number is the zero of the polynomial or not we have to find
( )
( )
( )
( )
( )
( )
( )
( )
And
Hence, x = 1, 2, 3 are the zeros of the polynomial p(x).
vTo check whether the given number is the zero of the polynomial or not we have to find
Here, , does not satisfy therefore, is not a zero of the polynomial.
viTo check whether the given number is the zero of the polynomial or not we have to find
Hence, x = 0 is the zeros of the polynomial.
vii To check whether the given number is the zero of the polynomial or not we have to find
Hence, is zeros of the polynomial.
viiiTo check whether the given number is the zero of the polynomial or not we have to find
Hence, does not satisyf the polynomial, so, is not zero of the polynomial.
Q u e s t i o n : 1 1
If x = 2 is a root of the polynomial f(x) = 2x
2
- 3x + 7a, find the value of a.
S o l u t i o n :
The given polynomial is
If x = 2 is the root of the polynomial.
Then
Q u e s t i o n : 1 2
If x = -
1
2
is a zero of the polynomial p(x) = 8x
3
- ax
2
-x + 2, find the value of a.
S o l u t i o n :
The given polynomial is
If is a zeros of the polynomial p(x).
then
Therefore,
Hence the value of
Q u e s t i o n : 1 3
If x = 0 and x = -1 are the roots of the polynomial fx
=2x
3
- 3x
2
+ ax + b, find the value of a and b.
S o l u t i o n :
The given polynomial is
fx
=2x
3
- 3x
2
+ ax + b
If is zeros of the polynomial f(x), then f
0 = 0
Similarly, if x = - 1 is the zeros of the polynomial of ,
Then,
Putting the value of b from equation
1
Thus,
Q u e s t i o n : 1 4
Find the integral roots of the polynomial fx
= x
3
+ 6x
2
+ 11x + 6.
S o l u t i o n :
The given polynomial is
Here, f(x) is a polynomial with integer coefficient and the coefficient of highest degree term is 1. So, the integer roots of f(x) are factors of 6. Which are by observing.
= 0
Also,
And similarly,
f
-3 = 0
Therefore, the integer roots of the polynomial f(x) are -1, -2, - 3
Q u e s t i o n : 1 5
Find rational roots of the polynomial fx
= 2x
3
+ x
2
- 7x - 6.
S o l u t i o n :
The given polynomial is
fx
is a cubic polynomial with integer coefficients. If
b
c
is rational root in lowest terms, then the values of b are limited
to the factors of 6 which are ±1, ±2, ±3, ±6
and the values of c are limited to the factor of 2 as ±1, ±2
. Hence, the possible
rational roots are ±1, ±2, ±3, ±6, ±
1
2
, ±
3
2
.
Since,
So, 2 is a root of the polynomial
Now, the polynomial can be written as,
Also,
Therefore,
Hence, the rational roots of the polynomial are 2, – 3/2 and – 1.
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FAQs on Factorization of Polynomials- 2 RD Sharma Solutions - Mathematics (Maths) Class 9
| 1. What is the concept of factorization of polynomials? |  |
Ans. Factorization of polynomials is the process of expressing a given polynomial as the product of its factors. These factors are usually lower degree polynomials. It helps in simplifying complex polynomials and solving equations involving polynomials.
| 2. How can we factorize a polynomial with a single variable? | |
Ans. To factorize a polynomial with a single variable, we need to identify its factors. This can be done by finding common factors, using the factor theorem, or applying different methods like grouping, trial and error, or synthetic division. The goal is to break down the polynomial into simpler terms that can be multiplied together to obtain the original polynomial.
| 3. What are the methods to factorize a quadratic polynomial? | |
Ans. Quadratic polynomials can be factorized using various methods. One common method is the "ac method" or "splitting the middle term," where we find two numbers that multiply to give the product of the coefficient of the quadratic term and the constant term, and also add up to the coefficient of the linear term. Another method is to use the quadratic formula to find the roots and then express the quadratic polynomial as a product of its linear factors.
| 4. Can all polynomials be factorized completely? | |
Ans. No, not all polynomials can be factorized completely. Some polynomials, especially those with irrational or complex roots, cannot be factorized into linear factors with real coefficients. In such cases, the polynomial is said to be irreducible over the set of real numbers. However, every polynomial can be factorized into linear and quadratic factors over the set of complex numbers.
| 5. How can factorization of polynomials be used in real-life applications? | |
Ans. Factorization of polynomials has several real-life applications. It is used in fields like physics and engineering to solve equations and model physical phenomena. In finance and economics, factorization is used in determining interest rates, analyzing investment options, and predicting market trends. It is also used in cryptography for secure communication and data encryption.
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Dec 28, 2024 Last updated |
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