GOOD MORNING
LET’S START OUR LESSON
MATH IS FUN!
ALGEBRAIC
MANIPULATION
( MANIPULASI ALJABAR )
Imagine we are required to design a
rectangular LCD monitor with variable
dimensions to serve different purpose. As the
dimensions vary, we may use algebraic
expressions to represents its length and
width. In this case, we can use length of the
𝑥+4
monitor is (2x+5) cm and the width is
cm.
2
( Bayangkan kita diharuskan merancang
monitor LCD persegi panjang dengan dimensi
variabel untuk melayani tujuan yang berbeda.
Karena dimensinya bervariasi, kita dapat
menggunakan
ekspresi
aljabar
untuk
menyatakan panjang dan lebarnya. Pada kasus
ini, kita menggunakan panjang monitor (2x +5)
𝑥+4
cm dan lebarnya
cm.)
2
ALGEBRAIC MANIPULATION
1.
2.
3.
4.
5.
6.
7.
8.
Expansion of Linier Algebraic Expressions.
Formulae for special products.
Factorisation of linear Algebraic Expressions of the Form ax+by
Factorisation of linear Algebraic Expressions of the Form ax+bx+kay+kby.
Factorisation of linear Algebraic Expressions of the Form 𝒂𝟐 𝒙𝟐 − 𝒃𝟐 𝒚𝟐
Factorisation of linear Algebraic Expressions of the Form 𝒂𝟐 ± 𝟐𝒂𝒃 + 𝒃𝟐
Factorisation of linear Algebraic Expressions of the Form 𝒙𝟐 + 𝒃𝒙 + 𝒄
Factorisation of linear Algebraic Expressions of the Form 𝒂𝒙𝟐 + 𝒃𝒙 +
𝒄, 𝒘𝒉𝒆𝒓𝒆 𝒂 ≠ 𝟏.
9. Multiplication and Division of Simple Algebraic Fractions.
Looking Back!
1. Recall that an algebraic expression is a combination of numbers,
variables ( or unknowns ) and operations signs. Circle the algebraic
expressions as show below. (Ingatlah bahwa ekspresi aljabar
adalah kombinasi angka, variabel (atau tidak diketahui) dan tanda
operasi. Lingkari ekspresi aljabar seperti yang ditunjukkan di
bawah ini. )
𝒙𝟐 − 𝟗
𝟒−𝟏×𝟐+𝟗
𝟕𝒙 + 𝟗𝒚
𝟐 + 𝟑(𝟏𝟏 − 𝟐)
Looking Back!
2. Identify each part of the expression 4x – 7. ( Identifikasi setiap
bagian dari ekspresi 4x – 7. )
x is (a) a variable
(b) a constant term
(c) a coefficient
-7 is (a) a variable
(b) a constant term
(c) a coefficient
4 is (a) a variable
(b) a constant term
(c) a coefficient
Looking Back!
3. Rewrite –(x-3) without the brackets. (Tulis ulang –(x-3) tanpa tanda
kurung.)
4. Simplify ( Sederhanakanlah : )
(a) −𝟑 𝟐 − 𝒙 + 𝟕
(b) 𝟖 − 𝟐 𝒙 + 𝟒 + 𝟑𝒙
(c) 𝟓 𝒙 + 𝟐 − 𝟒 𝒙 − 𝟑 − 𝟗
Expansion of Linier Algebraic Expressions.
Recall that in Secondary 1. we have learnt the distributive property of
multiplication over addition as follows.
𝒂 𝒃 + 𝒄 = 𝒂𝒃 + 𝒂𝒄
e.g. 𝟒 𝟕𝒙 − 𝟑 = 𝟒 𝟕𝒙 − 𝟒 𝟑
= 28x – 12
Note that this property can be expanded to any number of terms
inside the brackets.
𝒂 𝒃 + 𝒄 + 𝒅 + ⋯ + 𝒏 = 𝒂𝒃 + 𝒂𝒄 + 𝒂𝒅 + ⋯ + 𝒂𝒏
e.g. 5 (3x – 2y + 1 ) = 5(3x) + 5(-2y) + 5(1)
= 15x – 10y + 5
Expansion of Linier Algebraic Expressions.
Now, we are looking at the expansion of ( a + b )( c + d )
Before explaining how to do this expansion, let us consider 34 × 15.
• 34 × 15 = ( 30 + 4 ) ( 10 + 5 )
= (30 + 4 ) (10) + ( 30 + 4 ) (5)
= …..
= …..
Likewise, for the expansion of ( a + b )( c + d ), we treat ( a + b ) as a single term.
Using the distributive property, we have
𝒂+𝒃 𝒄+𝒅 = 𝒂+𝒃 𝒄 + 𝒂+𝒃 𝒅
= 𝒂+𝒃 𝒄 + 𝒂+𝒃 𝒅
= 𝒂𝒄 + 𝒃𝒄 + 𝒂𝒅 + 𝒃𝒅
FOIL METHOD
Consider the expression of
( a + b )( c + d )
We can apply the FOIL method
means,
First – Find the product of the
first terms.
Outer – Find the product of the
Outer terms.
Inner – Find the product of the
Inner terms.
Last – Find the product of the Last
terms.
Combining the four steps, we have
𝒂 + 𝒃 𝒄 + 𝒅 = 𝒂𝒄 + 𝒂𝒅 + 𝒃𝒄 + 𝒃𝒅