2.6 Factorisation
of Trinomials:
This concept is important
when we need to simplify algebraic expressions.
Do you know that 5-(3a^{2}-2a)( 6-3a^{2}+2a) = (3a+1)(a-1) (3a-5)(a+1) ? check the correctness
by substituting a=-1 and a=1. How do we prove that this equation holds good for all values of
a?
We have seen earlier that
HCF is useful in simplifying the algebraic expressions by taking this highest
common factor out side of of algebraic expressions.
The HCF of 4x^{2}y,
8x^{3}and 12xy
is 4x
_{} 4x^{2}y+8x^{3}+12xy
can be written as 4x (xy+2x^{2}+3y)
The process of writing an
algebraic expression as the product of two or more expressions (called factors) is called ‘factorisation’.
How
do we factorize trinomials of type x^{2}+mx +c?
Example
:
Let us take the expression x^{2}+x(a+b)+ab
x^{2}+x(a+b)+ab
= (x^{2}+xa)+(xb+ab) ( By rearranging
the terms)
= x(x+a)+b(x+a) ( x is common factor
of x^{2} and xa and b is common factor of xb
and ab)
= (x+a)(x+b)
We say x+a
and x+b are factors of the expression x^{2}+x(a+b)+ab
In other terms we say x^{2}+x(a+b)+ab
can be split in to product of x+a and x+b.
Example : x^{2}+5x+6 can be rewritten as
=x^{2}+3x+2x+6
=x(x+3)+2(x+3)
=(x+3)*(x+2)
Thus x+3 and x+2 are factors
of x^{2}+5x+6. They are of the form x+a and x+b.
These factors x+a and x+b of x^{2}+5x+6 are such that a+b=
5 and ab=6. By trial and error method we find that
a=3 and b=2 satisfy the condition a+b=5 and ab=6.
This is the reason why we
split 5x as 3x+2x and not as x+4x or anything else in the above example.
It is to be noted that not
all trinomials of type x^{2}+mx +c always have factors.
In later sections you will
learn the formula to find the factors for algebraic expression of type x^{2}+mx
+c.
x^{2}+5x+6
is of the form x^{2}+mx +c
with m = 5 and c=6
2.6 Problem 1: Factorise
x^{2}+27x+176
Solution:
We
need to find a and b such that a+b=27 and ab=176
The pairs of factors of 176
are (2, 88), (4, 44), (8, 22), (16, 11)
The –ve pairs of factors
(Ex (-2, -88)) are neglected as their sum cant be a
+ve number.
Of these pairs, we notice
that the pair (16, 11) satisfies the desired condition with a= 16 and b=11
So x^{2}+27x+176
can be rewritten as
x^{2}+16x+11x+
176
=x(x+16) +11(x+16)
=(x+16) (x+11)
Thus
(x+16) and (x+11) are factors of x^{2}+27x+176
Verification:
(x+16)(x+11) is of the form
(x+a)*(x+b) with a=16 and
b=11
_{}(x+16)*(x+11) = x^{2}+ x(16+11)+
16*11= x^{2}+27x+176 which is the given algebraic expression
2.6 Problem 2 :
Factorise x^{2}-6x-135
Solution:
We
need to find a and b such that a+b= -6 and ab= -135
The pairs of factors of
-135 are (3,-45), (-3, +45), (5,-27), (-5, +27), (9,-15), (-9, +15)
Of these pairs we notice
that 9-15 = -6 and 9*-15 = -135 satisfy the desired condition with a= 9 and b=
-15
_{}x^{2}-15x+9x -135
=x(x-15)+9(x-15)
=(x-15)(x+9)
Thus (x-15) and (x+9) are
factors of x^{2}-6x-135
Verification:
(x-15)(x+9)
is of the form (x+a)*(x+b)
with a=-15, b=9
_{}(x-15)*(x+9) = x^{2}+ x(-15+9)+ (-15*9)= x^{2}-6x-135 which is the given
algebraic expression
2.6 Problem 3: Factorise
m^{2}+4m-96
Solution:
We
need to find a and b such that a+b= 4 and ab= -96
The pairs of factors of -96
are (2,-48), (-2, 48), (3,-32), (-3, +32), (4,-24), (-4, +24), (6,-16), (-6,16), (8,-12), (-8,12)
Of these pairs we notice
that -8+12 = 4 and -8*12 = -96 satisfy the desired condition with a= -8 and
b=12
_{}m^{2}-8m+12m -96
=m(m-8)+12(m-8)
=(m-8)(m+12)
Thus (m-8) and (m+12) are
factors of m^{2}+4m-96
Verification:
(m-8)(m+12)
is of the form (m+a)*(m+b)
with a=-8, b=12
_{}(m-8)*(m+12) = m^{2}+ m(-8+12)+ -8*12= m^{2}+4m-96 which is the given
algebraic expression
Let us try to factorize the
expression of type px^{2}+mx +c (note the co-efficient of x^{2},
is p and not 1)
We need to find a and b
such that a+b=m and ab=pc
2.6 Problem 4 :
Factorize 24x^{2}-65x+21
Solution:
We
need to find a and b such that a+b= -65 and ab= 24*21 =504
The pairs of factors of
24*21 are(2,252), (-2,-252), (3, 138 ),
(-3,-138), (4,126), (-4,-126), (6,83),
(-6,-83), (8,63), (-8,-63), (9,56), (-9,-56), (12,42), (-12,-42)
Of these pairs we
notice that -9-56 = -65 and
-9*(-56) = 504=24*21
satisfies the desired condition with a=
-9 and b= -56
_{}24x^{2}-65x+21
=24x^{2}-9x -56x+21
( -65x is rewritten as -9x -56x)
=3x(8x-3)
-7(8x-3) (3x is common factor of 24x^{2} and 9x. -7 is common factor of
-56x and 21)
= (8x-3)(3x-7)
( 8x-3 is common factor )
Thus 8x-3 and 3x-7 are
factors of 24x^{2}-65x+21
Verification:
(8x-3)(3x-7)
=8x(3x-7)-3(3x-7) ( Multiply each of the terms)
=24x^{2}-56x
-9x+21 (simplification)
=24x^{2}-65x+21
which is the term given in the problem
2.6 Problem 5: Factorize 6p^{2}+11pq -10q^{2}
Solution:
We
need to find a and b such that a+b= 11 and ab= 6*(-10) =-60
The pairs of factors
of -60
are(2,-30), (-2,30),(3, -20 ),(-3,20) (4,-15), (-4,15),
(5,-12),(-5,12),(6,-10),(-6,10)
Of these pairs we
notice that -4+15 = 11 and -4*15 = -60 which satisfies our need of finding a and b
6p^{2}+11pq -10q^{2}
=6p^{2}+15pq -4pq-10q^{2}( 11pq = 15pq-4pq)
=3p(2p+5q)
-2q(2p+5q)
=(2p+5q)(3p-2q)
Thus 2p+5q and 3p-2q are factors of 6p^{2}+11pq -10q^{2}
Verification:
(2p+5q)(3p-2q)
=2p(3p-2q)+5q(3p-2q) ( Multiply each of the terms)
=6p^{2}-4pq
+15qp-10q^{2} (simplification)
= 6p^{2}+11pq -10q^{2}
which is the term given in the problem
2.6 Problem 6: Factorize
5-(3a^{2}-2a) (6-3a^{2}+2a)
Solution:
For easy working let x =3a^{2}-2a
Thus we need to factorize
5-x( 6-x)
5-x(
6-x)
= 5 -6x + x^{2}
= x^{2} -6x +5 = x^{2}
-5x -x+5
= x(x-5)-1(x-5)
= (x-1)(x-5)
By substituting value for x
we get
5-(3a^{2}-2a)( 6-3a^{2}+2a)
= (3a^{2}-2a -1) (3a^{2}-2a-5)
But 3a^{2}-2a -1 =
3a^{2}-3a+a -1 = 3a(a-1)+1(a-1) = (3a+1)(a-1)
3a^{2}-2a-5 = 3a^{2}+3a -5a-5 = 3a(a-1)-5(a+1) = (3a-5)(a+1)^{}
_{} 5-(3a^{2}-2a)( 6-3a^{2}+2a) = (3a+1)(a-1) (3a-5)(a+1)^{}
Verification:
1. Multiply
each of the individual terms to expand to check the correctness.
2. Check to be sure that the
answer is correct at least for
value of a=2 : (5 -8*-2) = 21 =
(7*1*1*3)
2.6
Summary of learning
No |
Points studied |
1 |
Finding
factors of x^{2}+mx +c such
that a+b=m and ab=c where x+a and x+b are its factors |
2 |
Split
px^{2}+mx +c such that a+b =m and ab=pc |