2. POLYNOMIALS (NCERT)
NCERT
2. POLYNOMIALS
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Chapter 2 : Polynomials |
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Exercise 2.1 Complete Solution Exercise 2.2 Complete Solution Exercise 2.3 Complete Solution Exercise 2.4 (Optional) Complete Solution |
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Important note : 1. A polynomial of degree one is called a linear polynomial.The general form of the linear polynomial is Note : The expression 2. A polynomial of degree two is called a quadratic polynomial. The general form of the quadratic polynomial is For example : 3. A polynomial of degree three is called a cubic polynomial. The general form of the cubic polynomial is For example : |
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Geometrical Meaning of the Zeroes of a Polynomial : Given a polynomial (i) A linear polynomial has only one zero . Because, the graph intersects the (ii) A quadratic polynomial has two zeroes . Because, the graph intersects the (iii) A cubic polynomial has three zeroes . Because, the graph intersects the |
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Relationship between Zeroes and Coefficients of a Polynomial : 1. For linear polynomial : If
The zero of the linear polynomial 2. For quadratic polynomial : If The sum of zeroes = The product of zeroes 3. For cubic polynomial : If The sum of zeroes The sum of the product of zeroes taken two at a time The product of zeroes |
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Division Algorithm for Polynomials : 4. Dividend = Divisor × Quotient + Remainder 5. If |
, where 
are real numbers . For example: 
, ………. etc.
etc. are not polynomials .
, where
and 
are real number .
……….etc.
, where 
and 
are real numbers.
,……… etc.
, the graph of 
intersects the 
, then 
is -ba= 
.
, then
and 
, then we can find polynomials 
and 
such that 
where 
or degree of 
degree of 
EXERCISE 2.1
1. The graph of 
are given in Fig. 2.10 below, for some polynomials 
. Find the number of zeroes of 
, in each case .
(i)
Solution : (i) The number of zeroes is 0 . Because, the graph does not intersect at the 
-axis .
(ii)
Solution : (ii) The number of zeroes is 1 as the graph intersects the -axis at one point only .
(iii)
Solution : (iii) The number of zeroes is 3 as the graph intersects the -axis at three points .
(iv)
Solution : (iv) The number of zeroes is 2 as the graph intersects the -axis at two points .
(v)
Solution : (v) The number of zeroes is 4 as the graph intersects the -axis at four points .
(vi)
Solution : (vi) The number of zeroes is 3 as the graph intersects the -axis at three points .
EXERCISE 2.2
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients .
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Solution :
(i) We have ,
Let , 
So, 
or 
The zeroes of are 
and 
.
The sum of its zeroes 
The product of its zeroes 
(ii) We have ,
Let , 
So, 
or 
The zeroes of 
are 
and 
.
The sum of its zeroes 
The product of its zeroes 
(iii) We have,
Let , 
So,
or 
The zeroes of are 
and 
.
The sum of its zeroes 
The product of its zeroes 
(iv) We have ,
Let , 
So, 
or 
The zeroes of 
are 0 and 2 .
The sum of its zeroes
The product of its zeroes 
(v) Let , 
So , 
or 
The zeroes of are 
and 
.
The sum of its zeroes 
The product of its zeroes 
(vi) We have ,
Let , 
So,
or 
The zeroes of are 
and .
The sum of its zeroes
The product of its zeroes
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively .
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Solution : (i)
Let 
and 
be zeroes of the quadratic polynomial 
respectively .
We have , 
and 
If 
, then 
and 
Therefore, the quadratic polynomial is 
i.e., 
.
OR
We know that ,
The quadratic polynomial
, where 
is any real constant .
Therefore, the quadratic polynomial is .
(ii)
Let and be zeroes of the quadratic polynomial respectively .
We have ,
and
If 
, then 
and 
Therefore, the quadratic polynomial is 
i.e., 
.
(iii)
Let and be zeroes of the quadratic polynomial respectively .
We have ,
and
If 
, then 
and 
Therefore, the quadratic polynomial is 
i.e., 
.
(iv)
Let and be zeroes of the quadratic polynomial respectively .
We have ,
and
If , then and
Therefore, the quadratic polynomial is 
i.e., 
.
(v)
Let and be zeroes of the quadratic polynomial respectively .
We have ,
and
If , then
and
Therefore, the quadratic polynomial is 
i.e., 
(vi)
Let and be zeroes of the quadratic polynomial respectively .
We have ,
and
If , then
and
Therefore, the quadratic polynomial is 
i.e., 
.
EXERCISE 2.3
1. Divide the polynomials by the polynomial 
and find the quotient and remainder in each of the following :
(i) 
, 
(ii) 
, 
(iii) 
, 
Solution : (i) We have,
,
Now ,
Therefore, the quotient is 
and the remainder is 
.
(ii) We have ,
,
Now ,
Therefore, the quotient is 
and the remainder is 8 .
(iii) We have,
,
Now ,
Therefore, the quotient is 
and the remainder is 
.
2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :
(i) 
(ii) 
(iii) 
Solution :
(i) We have,
Now ,
Therefore, the polynomial 
is a factor of the polynomial 
.
(ii) We have,
Now ,
Therefore, the polynomial 
is a factor of 
.
(iii) We have,
Now ,
Therefore , the polynomial 
is a factor of polynomial
.
3. Obtain all other zeroes of 
, if two of its zeroes are 
and 
.
Solution : Let 
Since two zeroes are and . So, 
is a factor of .
Now ,
; 
; 
or 
Therefore , the zeroes of the given polynomial are 
and -1
.
4. On dividing 
by a polynomial
, the quotient and remainder were 
and 
, respectively . Find .
Solution : Let , 
,
, 
Now ,
5. Give example of polynomials p(x) , g(x) , q(x) and r(x) , which satisfy the division algorithm and
(i) 
(ii) 
(iii) 
.
Solution : (i) Let 
; 
(ii) Let 
; 
; 
; 
(iii) Let 
; 
; 
; 
EXERCISE 2.4 (Optional)
1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case :
(i) 
(ii) 
Solution : (i) let , 
and 
and 
Again , 
Therefore, 
, 1 and 
are the zeroes of .
Let , 
and 
The sum of zeroes 
The sum the product of its zeroes taken two at a time 
The product of zeroes
Verified .
(ii) let , 
and 
and 1
Again , 
And
Therefore , 2 , 1 and 1 are the zeroes of .
Let , 
and 
The sum of zeroes 
The sum the product of its zeroes taken two at a time
The product of zeroes 
Verified .
2. Find a cubic polynomial with the sum , sum of the product of its zeroes take two at a time and the product of its zeroes as 
respectively .
Solution :
Solution: let , , and 
are the zeroes of the cubic polynomial 
respectively .
and 
If 
, then 
and 
So , the quadratic polynomial is 
3. If the zeroes of the polynomial 
and 
; find 
and 
.
Solution : Since, 
and
are the zeroes of the polynomial .
The sum of zeroes 
The product of zeroes = 
Therefore , the value of and
.
4. If two zeroes of the polynomial 
are 
, find other zeroes .
Solution: We have , 
Since two zeroes are 
and 
.
So , 
is the factor of .
Now ,
Quotient 
and Remainder 
The factor of 
are 
and 
.
So, its zeroes are 7 and – 5 .
Therefore, the zeroes of the given polynomial are , , 7 and – 5 .
5. If the polynomial 
is divided by another polynomial 
, the remainder comes out to be
, find 
and 
.
Solution : We have ,
Dividend 
Divisor 
Now ,
Remainder 
A/Q, 
Comparing the coefficients and constant terms on both the sides , we get
and 
[putting 
]
Therefore, the value of and 
.
Posted 4 years ago