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BINCY S BABY
B. Ed Mathematics
Reg. No : 18014350005
Of the angles and sides1
Of the sides and angles3
Chapter 2 Polynomials
Addition and subtraction.10
Multiplication and addition........11
Degree of a polynomial..............12
Meaning of remainder................13
OF ANGLES AND SIDES
Here a, b, c are the lengths of the sides of
a triangle and x, y, z are the lengths of the sides
of another triangle with the same angles.
We started with the triangles ABC and
XYZ in which A=X, B=Y, C=Z.
The numbers pairs (x, a), (y, b), (z, c) are
the lengths of the sides opposite the pairs of
is the number which shows how
many times the number x is the number a (or
what part of x is a).The other numbers
have similar meanings.
Whatever times (or part) we take to change
the side of length a to the side of length x,
such times(or part) is to be taken to change the
side of length b to the side of length y and the
side of length c to the side of length Z.
In other words, all these pairs of lengths
are in the same ratio. So, we can state this result
about the angles lengths of the sides of a
triangle with the same angles, as follows:
If all the angles of a triangle are equal to
the angles of another triangle, then all the pairs
of sides opposite equal angles have the same
OF SIDES AND ANGLES
Does the pair of triangles with proportional
sides also have the same angles?
Lets consider the triangle shown below
We want to check whether the triangles
have the same angles.
For that, what we do is draw a third triangle
with the same angles of ABC, and which is
congruent to XYZ.
OR = x, Q=B, R=C
So, we must also have P=A. Thus the angles
of PQR are equal to the angle of ABC.
So, by the theorem we have proved,
We actually draw PQR with QR = x, So,
We started with
Comparing the last two questions, we get
From these, we get
RP = y and PQ = z
Thus the three sides of PQR are equal to the
three sides of XYZ and so these triangles are
congruent. So, the angles opposite their pairs of
equal sides are also equal:
X=P, Y=Q, Z=R
We have already seen that
P=A, Q=B, R=C
Thus we have
X=A, Y=B, Z=C
How do we state the result just proved?
If the sides of a triangle are proportional to the
sides of another triangle, then the angles
opposite such sides are equal
If the angles of a triangle are all equal to the
angles of another triangle, then the sides of the
two angles are proportional: and on the
otherhand, if the lengths of the sides of another
triangle, then the angles of one triangle are equal
to the angles of the other.
Two triangles related in this way are said to
We have seen two ways to show that two
triangles are similar: either proves that they have
the same angles or prove that their sides are
There is a third way; if a pair of sides is
proportional and their included angles are equal,
then also the triangles are similar.
if two sides of a triangle are proportional
to two sides of another triangle and if their
included angles are equal, then the triangles are
Various positive integral powers of x are
multiplied by specific numbers (if may be by 1
also), the products are added or subtracted and
finally a specific number is added or subtracted
(it may be 0) such algebraic expression are
A polynomial is usually written with the
exponents of x in descending order, the number
without x at the end.
Thus for example, 35x-24x2+4x3 is usually
written as 4x3-24x2+35x; and 15+3x as 3x+15.
Look at 2x3-3x2+x-5. We can write it as
Each such addend is called a term of the
Thus in the polynomial we considered just
now, the terms are 2x3, -3x2, x, -5.
Now lets look at the terms themselves.
Apart from the last term, all others are got by
multiplying (Positive integral) powers of x by
specific numbers. Each such multiplier is called
the Co-efficient of the corresponding power.
Thus in our example, the coefficient of x3
is 2, the coefficient of x2 is -3 and the coefficient
of x is 1.
Thus the number added at the end is called
the constant term.
ADDITION AND SUBTRACTION
Lets simplify (2x+3) + (4x+6)
(2x+3) + (4x+6) = 2x+3 + 4x+6
= (2x+ 4x) + (3 +6)
(2x +3) - (4x+6) = 2x+3 - 4x-6
= (2x- 4x)+(3 -6)
= - 2x-3
= (4+2) x2+ (3+5) x+(1-2)
(4x2+3x+1) - (2x2+5x-2)
= (4 - 2)x2+(3 - 5)x+(1 + 2)
= 2x2 - 2x + 3
(x+y) (u+v) = xu+xv+yu+yv
ie, Each number in the first sum should be
multiplied by each number in the second sum
and all these products should be added.
= (2x4x) + (2x 3) + (54x) + (53)
MULTIPLICATION AND ADDITION
We can see that for any 3 polynomials
p(x), q(x), r(x).
p(x)r(x)+q(x)r(x) = (p(x)+q(x))r(x)
=((2x+3)+( x+4)) (x+1)
= (3x+7) (x+1)
DEGREE OF POLYNOMIAL
The polynomial P(x) =6x4+4x3+5x+1.The
power of x in it are x4, x3, x. The largest
exponent among them is 4. It is called the
degree of the polynomial.
A polynomial whose degree is 1 is called a
first degree polynomial. A polynomial whose
degree is 2 is called a second degree polynomial
and so on.
MEANING OF REMAINDER
In general, when a polynomial cannot be
completely divided by another polynomial, the
remainder should be a polynomial of degree less
than that of the divisor, or a number.
We want to find the quotient and reminder
when 4x2+5 is divided by 2x+3.As taking the
quotient as ax+b and the remainder as c, we get
(ax+b)(2x+3)+c = 4x2+5
ie, 2ax2+(3a+2b)x+(3b+c) = 4x2+5
Therefore, 2a= 4, 3a+2b=0, 3b+c=5
ie, 2a=4 ; a=2
For a=2 and 3a+2b=0.We must have 2b= -3a =-6
ie, b= -3 and for b= -3 and 3b+c=5,
Thus the quotient is (2x-3) and the remainder 14