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Algebra
Clement
Radcliffe, Contributor
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| Students
from Papine High School view the
World Town Planning Day exhibition
at the Jamaica Conference Centre
on Tuesday, November 8. - Junior
Dowie Photo |
AS
WE continue to review Algebra, I wish
to remind you of the following:
*
The concepts included in Algebra are
fairly routine and, with effort, you
all should be able to do them well.
*
Many areas were done in the lower
forms and must be effectively revised.
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Algebra should be selected as one
of the compulsory topics in Section
2.
We
will now review last week's homework.
We
will now continue algebra with the
topic factorisation.
Note
that an algebraic expression is factorized
when it is expressed as the product
of its simplest factors. The usual
methods are:
(a)
Common factor
(b)
Grouping
(c)
Factorizing of quadratic expressions
(d)
Difference of two squares.
The
methods are adequately explained in
the text books and you should use
them to aid you as you revise for
your exams.
It
is important that you do the following
in all cases:
(a)
Bring each factor to its simplest
form, for example, a factor 3x + 9
should be expressed as 3(x + 3)
(b)
Check your answers, if you have the
time, by expanding and comparing the
result with the original expression.
The
following examples are presented for
your benefit.
EXAMPLES
OF COMMON FACTOR METHOD
1.
Factorize 8x²
- 12x
The
common factor method is used, as 4x
is the factor which is common to both
terms. Both terms are divided by 4x
for us to obtain the second factor.
Answer:
4x(2x - 3)
2.
Factorize 15x²y
-10xy³
Note
that the common factor is 5xy
ie.
Answer is 5xy(3x - 2y2)
EXAMPLES
OF GROUPING METHOD
3.
Factorize ax + ay + bx + by
Note
that a is the common factor of ax
+ ay and b the common factor of bx
+ by
ie.
ax + ay + bx + by = a(x + y) + b(x
+ y)
Do
you realise that (x + y) is common
to both expressions?
ie.
a(x + y) + b(x + y) = (x + y)(a +
b)
This
method could therefore be described
as repeated common factor method.
4.
Factorize 2ax - 6ay + bx - 3by
2a(x
- 3y) + b (x - 3y)
=
(x - 3y)(2a+ b)
EXAMPLES
OF METHOD OF FACTORIZING QUADRATIC
EXPRESSIONS
5.
Factorize x2 + 8x + 15
This
method is based on the principle that
(x + b) (x + c) = x2 + (b + c) x +
bc. Do you see a relationship between
(b + c) which is the coefficient of
x, bc which is the constant term,
and b and c which are the values in
the brackets on the left hand side?
This relationship and the "trial
and error" plays an important
role in this method.
Using
the above:
x²
+ 8x + 15 = (x + 5)(x +3)
If
you have not realised the relationship
mentioned above, then please note
that:
*
5 + 3 = 8 (coefficient of x)
*
5 x 3 = 15 (The constant term)
You
may use "trial and error"
to identify 5 and 3, the values which
satisfy the relationship.
6.
Factorize 2x²
+5 x 12
Despite
the coefficient of x2 being 2, a method
similar to that of example 5 above
is used.
...
2x²
+ 5x - 12 = (2x - 3)(x + 4)
EXAMPLES
OF METHOD OF DIFFERENCE OF TWO SQUARES
7.
Factorize 9x²
- 4
This
is based on the fact that a2 - b2
= (a - b)(a + b). The critical problem
is therefore to find the square root
of each term.
We
will try another example.
8.
Factorize 1- (a + b) 2
By
factorizing, then you can show that
1-
(a + b)²
= [1 - (a + b)] [1 + (a + b)]
=
(1 - a + b)(1 + a + b)
Competence
is developed in the solution of these
problems if you practise extensively.
Remember to check your answers by
expanding the factors.
Now,
please attempt the following.
Factorize
(a)
9a²
- b²
(b)
3x -8y - 4xy + 6
(c)
x²
- y²
- 4x + 4y
(d)
16/x²
-1
(e)
3x²
- 7x -6
*
Clement Radcliffe is principal
of Glenmuir High School in Clarendon.
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