|
Algebra
(cont'd)
Clement
Radcliffe, Contributor
 | Students
in performance at Denham Town High School's career day expo on Thursday, May 18.
- Ricardo Makyn/Staff Photographer |
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.
- Algebra
should be selected as one of the compulsory topics in Section 2.
We
will now review last week's homework. 1.
Solve 2(x + 2y) - 4(x - y) = (A)
- 2x + 8y (B)- 2x (C) 6x + 8y (D) - 2x + y Solution
- 2(x
+ 2y) - 4(x - y). Clearing the brackets:
2x
+ 4y - 4x + 4y = - 2x + 8y
The
answer is (A).
2.
Solve (5P - 2) - 3(P + 4) = 8 Solution
(5P
- 2) - 3(P + 4) = 8 Clearing the brackets: 5P
- 2 - 3P - 12 = 8.
Therefore, 2P = 8 + 14 = 22
Therefore, P = 11 | 3.
Solve | 5y
- 4 | - | 3y
- 7 | =
y | | | 4 | | 2 | |
You
are reminded that you may: - Simplify
the left-hand side and then equate it to y.
- Multiply
both sides by 4 which is the L.C.M. of 4 and 2.
In
this case, the latter is recommended. | Therefore,
4 x | 5y
- 4 | -
4 x | 3y
- 7 | 4
x = y | | | 4 | | 2 | |
Therefore,
5y - 4 - 2(3y - 7) = 4y
Therefore, 5y - 4 - 6y + 14 = 4y
Therefore, -5y
= -10
Therefore, y = -10/-5 =2
Answer
is y = 2. We
will now continue algebra with the topic Factorisation. Note
that an algebraic expression is factored when it is expressed as the product of
its simplest factors. The usual methods are: (a)
Common factor
(b)
Grouping
(c)
Factorising of Quadratic Expressions
(d)
Difference of two squares. The
methods are adequately explained in the textbooks 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. COMMON
FACTOR METHOD 1.
Factorise: 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.
Factorise: 15x²y
-10xy³ Note
that the common factor is 5xy Therefore,
the answer is 5xy(3x - 2y²)
GROUPING
METHOD 3.
Factories ax + ay + bx + by Note
that a is the common factor of ax + ay and b the common factor of bx + by Therefore,
ax + ay + bx + by = a(x + y) + b(x + y) Do
you realise that (x + y) is common to both expressions? Therefore,
a(x + y) + b(x + y) = (x + y)(a + b) This
method could, therefore, be described as repeated common factor method.
4.
Factorise 2ax - 6ay + bx - 3by
2a(x
- 3y) + b (x - 3y)
=
(x - 3y)(2a+ b) METHOD
OF FACTORISING QUADRATIC EXPRESSIONS 5.
Factories x²
+ 8x + 15 This
method is based on the principle that (x + b)(x + c) = x? + (b + c) x + bc. Do
you see the 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.
Factorise: 2x²
+ 5x -12 Despite
the coefficient of x²
being 2, a method similar to that of example 5 above is used. Therefore,
2x² +
5x - 12 = (2x - 3)(x + 4) DIFFERENCE
OF TWO SQUARES 7.
Factorise: 4 - x² This
is based on the fact that a? - b? = (a - b)(a + b). The critical problem is therefore
to find the square root of each term. As
 4 = 2 and x2
= x Therefore
4 - x²
= (2 - x)( 2 + x). We
will try another example. 8.
Factorise: 9x²
- 16 By
using the method of difference of two squares you can show that since 9x²
= 3x and 16 = 4, then
9x²
- 16 = (3x - 4)(3x + 4). 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 for homework. Factorise
(a)
9a² -
b² (d)
3x -8y - 4xy + 6
(b)
x² - y²
- 4x + 4y (e) 16x²
-1
(c) 3x²
- 7x -6
Clement
Radcliffe teaches at Glenmuir High School. | 
