TOPIC : Polynomials
1. Introduction
2. Geometrical Meaning of the Zeroes of a Polynomial
3. Relationship between Zeroes and Coefficients of a Polynomial
4. Division Algorithm for Polynomials
1. Finding number of zeroes by observing the graf of the polynomials.
2. Finding zeroes of any polynomial and verifying the relationship between the
zeroes and the coefficients of the polynomial.
3. Dividing any polynomial by another polynomial and verifying the division
algorithm of polynomials.
4. Applying remainder theorem and factor theorem .
1.
Understanding the geometric meaning of the zeroes of the polynomials.
2. Able to find zeroes of the polynomial and veryfication the relation between the
zeroes and the coefficients of the polynomial.
EXAMPLES :
TOPIC : PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
CONCEPT :
Recall, from Class IX, that the following are examples of linear equations in two variables.
Consider the following situation:
In a One-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian
batsmen together scored 176 runs. Express this information in the form of an equation.
Here, you can see that the score of neither of them is known, i.e., there are two unknown quantities. Let us
use x and y to denote them. So, the number of runs scored by one of the batsmen is x, and the number of
runs scored by the other is y. We know that
x + y = 176,
which is the required equation.
This is an example of a linear equation in two variables. It is customary to denote the variables in
such
equations by x and y, but other letters may also be used. Some examples of linear equations in
two
variables are:
1.2s + 3t = 5, p + 4q = 7, πu + 5v = 9 and 3 = 2 x – 7y.
Note that you can put these equations in the form 1.2s + 3t – 5 = 0, p + 4q – 7 = 0, πu + 5v – 9 = 0 and
2 x – 7y – 3 = 0, respectively.
So, any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a
and b are not both zero, is called a linear equation in two variables. This means that you can think
of
many many such equations. We also know about the solutions of a linear equation which are infinite.
Pair of Linear Equations in Two Variables
Now consider another example :
Ragini went to a stationery shop and purchased 2 pencils and 3 erasers for Rs 9. Her friend Soniya saw
the new variety of pencils and erasers with Ragini, and she also bought 4 pencils and 6 erasers of
the
same kind for Rs 18. Represent this situation algebraically.
Let us denote the cost of 1 pencil by Rs x and one eraser by Rs y. Then the
algebraic representation is given by the following equations:
2x + 3y = 9
4x + 6y = 18
These two linear equations are in the same two variables x and y. Equations like these are called a pair
of linear equations in two variables.
Let us see what such pairs look like algebraically.
The general form for a pair of linear equations in two variables x and y is
a1x + b1y + c1 = 0
and a2x + b2 y + c2 = 0,
where a1, b1, c1, a2, b2, c2 are all real numbers and a12 + b12 ≠0, a22 + b22 ≠ 0.
Some examples of pair of linear equations in two variables are:
2x + 3y – 7 = 0 and 9x – 2y + 8 = 0
5x = y and –7x + 2y + 3 = 0
x + y = 7 and 17 = y
GEOMETRICAL REPRESENTATION
Recall, that we have studied in Class IX that the geometrical (i.e., graphical) representation of a linear
equation in two variables is a straight line. Can you now suggest what a pair of linear equations in two
variables will look like, geometrically? There will be two straight lines, both to be considered together.
You have also studied in Class IX that given two lines in a plane, only one of the following
three
possibilities can happen:
(i)
The two lines will intersect at one point.
(ii)
The two lines will not intersect, i.e., they are parallel.
(iii)
The two lines will be coincident.
Graphical Method of Solution of a Pair of Linear Equations
In the previous class, you have seen how we can graphically represent a linear equation. Similarly we can
draw the graph for pair of linear equations .You have also seen that the lines may intersect, or may
be parallel, or may coincide. Now we solve them in each case if possible..
Let us look some examples.
(1) x – 2y = 0 (i)
and
3x + 4y = 20 (ii)
Let us represent these equations graphically. For this, we need at least two solutions for each
equation. We give these solutions in Table.
Plot the points A(0, 0), B(2, 1) and P(0, 5), Q(4, 2), corresponding to the solutions in Table. Now
draw the lines AB and PQ, representing the equations x – 2y = 0 and 3x + 4y = 20, as shown .
Since both the lines intersect each other at point (4, 2). Therefore x = 4, y = 2 is a solution of both the
equations.
Since (4, 2) is the only common point on both the lines, there is one and only one solution for this
pair of linear equations in two variables.
(2)
2x + 3y = 9 (i)
4x + 6y = 18 (ii)
We plot these points in a graph paper and draw the lines. We find that both the lines coincide .This is so,
because, both the equations are equivalent, i.e., one can be derived from the other The solutions of
the
equations are given by the common points. Are there any common points on these lines? From the graph,
we observe that every point on the line is a common solution to both the equations. So, the
equations
2x + 3y = 9 and 4x + 6y = 18 have infinitely many solutions. This should not surprise us, because if we
divide the equation 4x + 6y = 18 by 2 , we get 2x + 3y = 9, which is the same as Equation (1). That is,
both the equations are equivalent i.e., one can be derived from the other.
(3)
x + 2y – 4 = 0
2x + 4y – 12 = 0
(i)
(ii)
To represent the equations graphically, we plot the points R(0, 2) and S(4, 0), to get the line RS and the
points P(0, 3) and Q(6, 0) to get the line PQ.
We observe in Fig, that the lines do not intersect anywhere, i.e., they are parallel.
So, we have seen several situations which can be represented by a pair of linear equations. We
have seen their algebraic and geometric representations.
HOME WORK
1. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then.
Also, three years from now, I shall be three times as old as you will be.” (Isn’t this
interesting?) Represent this situation algebraically and graphically and solve them.
2. The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another
bat and 2 more balls of the same kind for Rs 1300. Represent this situation algebraically
and geometrically and solve them .
3. The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs 160. After a
month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation
algebraically and geometrically and solve them.