Polynomial in Two Variable
In last chapter we have learnt about polynomial in
one variable. In that chapter we have learnt
Definition of Polynomial :- The algebraic expression that is formed by variable and
constant, is known as Polynomial.
It may be of order one, two .three or so on
For Example X^2 + 4X +3 (One variable Polynomial)
Variable :-
The variable are changing it’s value according to the sum (Questions) or
situation.
For example :- Months in a year
As soon as the year going the month changes like as Jan, Feb, Mar,
….so on.
In this situation the month changes from Jan to Dec. So it is
called variable.
It may be x, y, z, a, b, c ……
Constant :-
It is part of polynomial that never changes it’s value according to situation
or sum (Questions)
For example :- 1, 2, 3, 4, …..
In brief we can say all negative integers and whole numbers are
constant.
Types of polynomials
It may be categories in following types
1. Polynomial of one Variable :- In this only one variable take place in
algebraic expression. Like x or y or z…. so on
For Example 4 X^2 + 5 X +7 (In this only
x changes it’s value so it is polynomial of one variable)
2. Polynomial of two Variable :- In this two variable take place in
algebraic expression. It is a combination of two variables like as (x and y) or
(y and z) or (z and x)
For examples :- (X^2) * Y + 4 * X * (Y^2) -7 (In
this only x and y changes it’s value so it is polynomial of two variable)
Degree of Polynomial of Two Variable:-
The highest power of any algebraic expressions is known as Degree
of polynomial.
For example (4 x^2 * y + 5 x * y^2 +7)
In this given expression the highest power of variable x is 2 in
first term (4 x^2 * y), so the degree of this polynomial is 2 for first term (4
x^2 * y) and the highest power of variable x is 1 in second term (5 x * y), so
the degree of this polynomial is 1 for second term (5 x * y ) .
The
highest power of variable y is 1 in first term (4 x^2 * y), so the degree of
this polynomial is 1 for first term (4 x^2 * y) and the highest power of
variable y is 2 in second term (5 x * y^2), so the degree of this polynomial is
2 for second term (5 x * y^2 ).
If we
are talking about constant term so in this case degree of x and y is
0 Like as third term 7 because in this there are no power of x or y present so
for degree of x and y is zero.
According to number of terms it can
be categories in following types :-
1. Monomial :- The
polynomial having number of term one, known as ‘Monomial’
For example :- (9 x^2 * y), (7 y^3
* z)
2. Binomial :- The polynomial
having number of terms two, known as ‘Binomial’
For example :- (x^2 * y + x * y^2),
(z * y^2 + y * z^2 )
3. Trinomial :- The polynomial
having number of terms three, known as ‘Trinomial’
For example :- x^3 + x^2 + x + 3,
y^3 + y^2 + y + 7
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| maths ncert solutions |
Problems :-
And the cost of ball = y
So equation may be
x = 10 y
x – 10 y = 0
2. If a cost of cooler is twice, is
the cost of fan. Write the equation for that ?
Sol. Let the cost of cooler = x
And the cost of fan = y
So equation may be
x = 2 y
x – 2 y = 0
3. Find the value of a, b, and c
for given polynomial ?
a). 4 x + 7 y = 9
b). 7 x + 9 y = 11
c). ½ x + y = 2
d). 2 x + ½ y + 3 = 0
e). 3 x + 2 y = 0
a). Given 4 x + 7 y
= 9
4 x + 7 y – 9 = 0
Compare this equation ( a x + b y +
c = 0 )
a = 4 b = 7
c = -9
b). Given 7 x + 9 y
= 11
7 x +
9 y – 11 = 0
Compare this equation ( a x + b y + c = 0
)
a = 7
b = 9 c = -11
c). Given 1/2 x + y = 2
1/2 x + y – 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = 1/2
b = 1 c = -2
d). Given 2 x + 1/2
y + 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = 2
b = 1/2 c = 3
e). Given 3 x + 2 y =
0
Compare this equation ( a x + b y + c = 0
)
a = 3
b = 2 c = 0
4. Find the value of a, b, and c
for given polynomial ?
a). 4 x - 7 y = 9
b). 7 x - 9 y = 11
c). ½ x - y = 2
d). 2 x - ½ y + 3 = 0
e). 3 x - 2 y = 0
a). Given 4 x - 7 y
= 9
4 x - 7 y – 9 = 0
Compare this equation ( a x + b y +
c = 0 )
a = 4 b = -7
c = -9
b). Given 7 x - 9 y
= 11
7 x -
9 y – 11 = 0
Compare this equation ( a x + b y + c = 0
)
a = 7
b = -9 c = -11
c). Given 1/2 x - y = 2
1/2 x - y – 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = 1/2
b = -1 c = -2
d). Given 2 x - 1/2
y + 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = 2
b = -1/2 c = 3
e). Given 3 x - 2 y =
0
Compare this equation ( a x + b y +
c = 0 )
a = 3 b = -2
c = 0
5. Find the value of a, b, and c
for given polynomial ?
a). -4 x - 7 y = 9
b). -7 x - 9 y = 11
c). -½ x - y = 2
d). -2 x - ½ y + 3 = 0
e). -3 x - 2 y = 0
a). Given -4 x - 7 y
= 9
-4 x -
7 y – 9 = 0
Compare this equation ( a x + b y + c = 0
)
a = -4
b = -7 c = -9
b). Given -7 x - 9 y
= 11
-7 x -
9 y – 11 = 0
Compare this equation ( a x + b y + c = 0
)
a = -7
b = -9 c = -11
c). Given -1/2 x - y = 2
-1/2 x - y – 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = -1/2
b = -1 c = -2
d). Given -2 x - 1/2
y + 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = -2
b = -1/2 c = 3
e). Given -3 x - 2 y =
0
Compare this equation ( a x + b y +
c = 0 )
a = -3 b = -2
c = 0
6. Find the value of a, b, and c
for given polynomial ?
a). -4 x - 7 y = -9
b). -7 x - 9 y = -11
c). -½ x - y = -2
d). -2 x - ½ y - 3 = 0
a). Given -4 x - 7 y
= -9
-4 x -
7 y + 9 = 0
Compare this equation ( a x + b y + c = 0
)
a = -4
b = -7 c = 9
b). Given -7 x - 9 y
= -11
-7 x -
9 y + 11 = 0
Compare this equation ( a x + b y + c = 0
)
a = -7
b = -9 c = 11
c). Given -1/2 x - y = -2
-1/2 x - y + 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = -1/2
b = -1 c = 2
d). Given -2 x - 1/2
y - 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = -2
b = -1/2 c = -3
7. Find the value of a, b, and c
for given polynomial ?
a). x = 9 y
b). x = 2 y
c). 3 x = 2
d). ½ y + 3 = 0
e). 2 y = 0
a). Given x = 9 y
x
- 9 y = 0
Compare this equation ( a x + b y + c = 0
)
a = 1
b = -9 c = 0
b). Given x = 2 y
x - 2 y = 0
Compare this equation ( a x + b y + c = 0
)
a = 1
b = -2 c = 0
c). Given 3 x = 2
3 x + 0 y – 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = 3
b = 0 c = -2
d). Given 1/2 y + 3 = 0
0 x + 1/2 y – 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = 0
b = 1/2 c = -3
e). Given 2 y = 0
0 x + 2 y = 0
Compare this equation ( a x + b y +
c = 0 )
a = 0 b = 2
c = 0
8. Find the value of a, b, and c
for given polynomial ?
a). x = -9 y
b). x = -2 y
c). 3 x = -2
d). ½ y - 3 = 0
e). -2 y = 0
a). Given x = -9 y
x +
9 y = 0
Compare this equation ( a x + b y + c = 0
)
a = 1
b = 9 c = 0
b). Given x = -2 y
x + 2 y = 0
Compare this equation ( a x + b y + c = 0
)
a = 1
b = 2 c = 0
c). Given 3 x = -2
3 x + 0 y + 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = 3
b = 0 c = 2
d). Given 1/2 y - 3 = 0
0 x + 1/2 y + 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = 0
b = 1/2 c = 3
e). Given -2 y = 0
0 x - 2 y = 0
Compare this equation ( a x + b y +
c = 0 )
a = 0 b = -2
c = 0
9. Find the value of a, b, and c
for given polynomial ?
a). -x = 9 y
b). -x = 2 y
c). -3 x = 2
d). -½ y - 3 = 0
e). -3 y = 0
a). Given -x = 9 y
-x -
9 y = 0
Compare this equation ( a x + b y + c = 0
)
a = -1
b = -9 c = 0
b). Given -x = 2 y
-x - 2 y = 0
Compare this equation ( a x + b y + c = 0
)
a = -1
b = -2 c = 0
c). Given -3 x = 2
-3 x + 0 y - 2 = 0
Compare this equation ( a x + b y + c = 0
)
a = -3
b = 0 c = -2
d). Given -1/2 y - 3 = 0
0 x - 1/2 y - 3 = 0
Compare this equation ( a x + b y + c = 0
)
a = 0
b = -1/2 c = -3
e). Given -3 y = 0
0 x - 3 y = 0
Compare this equation ( a x + b y +
c = 0 )
a = 0 b = -3
c = 0
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