FACTORING ALGEBRAIC EXPRESSIONS

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1 FACTORING ALGEBRAIC EXPRESSIONS
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2 P Q P + Q P – Q P x Q FACTORING ALGEBRAIC EXPRESSIONS
I. Arithmetical operations on algebraic : Adding Substracting Multiplying Dividing Exponentiating P Q P + Q P – Q P x Q 2x – 3 4x + 1 6x -2 -2x – 4 8x x – 3 x + 2 2x – 1 3x + 1 -x + 3 2x2 + 3x – 2 2a – a2 2a 4a – a2 -a2 4a2 – 2a3 2(3 – a) 1 – a 7 – 3a 5 – a 6 – 8a + 2a2 4b - 1 (2b + 1) 6b 2b – 2 8b2 + 2b – 1

3 P Q P : Q P P2 P3 4a 2 2a 4a2 6a3 3a 2a2 12a – 4a2 6 – 2a 8a + 16 4
-3ab3 9a2b6 -27a3b9 4ab 16a2b2 64a3b3

4 II. Factoring Algebraic Expressions
12 ab2 – 9b3c2 = 3b2 (4a – 3bc2) 4x = 22.x 12ab2 = 22.3.a.b2 8 = 23 9b3c2 = 32.b3.c2 GCD = 22 = 4 GCD = 3.b2 Exercise 12xy2 + 4x2y3z 8x – 12y 24xyz2 + 9x2y 10xy3 + 2y2z 1. 4xy2 (3 + xyz) 2. 4 (2x - 3y) 3. 3xy (8z2 + 3x) 4. 2y2 (5xy + z)

5 III. Special form A. x2 + 2xy + y2 = (x + y) 2 B. x2 - 2xy + y2 = (x - y) 2 (x + y) 2 =(x + y)(x + y) (x - y) 2 =(x - y)(x - y) = x2 + xy + xy + y2 = x2 - xy - xy + y2 = x2 + 2xy + y2 = x2 - 2xy + y2 x2 + 4x + 4 = ( … ) 2 ( x + 2) 2 x2 - 6x + 9 = ( … ) 2 ( x - 3) 2

6 A. x2 + 2xy + y2 = (x + y) 2 B. x2 - 2xy + y2 = (x - y) 2 a2 + 4a + 4
Factor the following algebraic expressions! a2 + 4a + 4 16x2 – 24x + 9 4a2 – 4ab + b2 9m2 + 30mn + 25n2 25p2 + 70pq + 49q2 1. (a + 2)2 2. (4x - 3)2 3. (2a - b)2 4. (3m + 5n)2 5. (5p + 7q)2

7 a2 – b2 (2m) 2 – 32 x2 – 49 m2 – 121 64 – y2 1. (a – b)(a + b)
C. x2 - y2 C. = (x - y)(x + y) (x – y )(x + y) = x2 + xy – xy - y2 = x2 - y2 a2 – b2 (2m) 2 – 32 x2 – 49 m2 – 121 64 – y2 1. (a – b)(a + b) 2. (2m – 3)(2m + 3) 3. (x – 7 )(x + 7) 4. (m – 11 )(m + 11) 5. (8 – y)(8 + y)

8 D. Factoring ax2 + bx + c, when a = 1
x2 + bx + c = (x + p)(x + q) Where c = p x q and b = p + q Example : 1. x2 + 10x + 16 ====> a = 1, b = 10, c = 16 p = …? q = …? … x … = 16 … + … = 10 8 2 8 2 x2 + 10x + 16  (x + 8)(x + 2) 2. x2 – x – 6 ====> a = 1, b = -1, c = -6 p = …? q = …? -3 … x … = -6 … + … = -1 2 x2 – x – 6  (x – 3)(x + 2) -3 2

9 a2 + 5a + 6 a2 + a – 6 (a + 3)(a + 2) (a – 2 )(a + 3) y2 + 6y + 9
Factor the following algebraic expressions! a2 + 5a + 6 a2 + a – 6 y2 + 6y + 9 y2 – 14y + 24 p2 + 4p – 5 (a + 3)(a + 2) (a – 2 )(a + 3) (y + 3)(y + 3) (y – 12)(y – 2) (p + 5)(p – 1)

10 E. Factoring ax2 + bx + c, when a ≠ 1
p + q = b p x q = a x c Example : 1. 2x2 + 7x + 6 ===> a = 2, b = 7, c = 6 p = …? q = …? … + … = 7 … x … = 12 4 3 4 3 2x2 + 7x + 6  2x2 + 4x + 3x + 6  2x2 + 4x + 3x + 6  2x(x + 2) + 3(x + 2)  (x + 2) (… + …) (2x + 3)

11 2. 6x2 + 13x - 5 ===> a = 6, b = 13, c = -5 p = …? q = …? … + … = 13 … x … = -30 -2 15 -2 15 6x2 + 13x - 5  6x2 - 2x + 15x - 5  6x2 - 2x + 15x - 5  2x(3x - 1) + 5(3x - 1)  (3x - 1) (… + …) (2x + 5)

12 IV. OPERATIONS ON ALGEBRAIC FRACTIONS
Example:

13 LINES 1. SLOPE 2. GRAPHING LINEAR EQUATIONS 3. WRITINGLINEAR EQUATIONS
4. PARALLEL & PERPENDICULAR Created by ﺠﻴﻄ for mathlabsky.wordpress.com Created by ﺠﻴﻄ for mathlabsky.wordpress.com

14 Slope of a Line Slope (gradient) is a ratio of the change in y (vertical change) to the change in x ( horizontal change) The slope, denoted by m, of the line through the points and Is defined as follows: Y ● ● Invers X

15 Find the slope of each segment (a, b, c and d)!
 3 -5 6 -3 a b c d Find the slope of each segment (a, b, c and d)! Lines with positive slope rise to the right Lines with negative slope fall to the right

16 k h Find the slope of line k and h! Slope of line 4 Slope of line 6 4
8

17 Linear equations can be written in different forms :
Standard form and slope-intercept form. Form Equation Slope Standard Slope-intercept Example : Equation Form Slope standard standard Slope-intercept Slope-intercept standard Slope-intercept HOME

18 Y X Graphing Linear Equations
To draw the line we need two point determine a line. We can find the X-intercept and Y-intercept. Y X Example : graph the line 2x + 3y = 12 To find X-intercept, let y = 0 ● (0 , 4) Thus, (6 , 0) is a point on the line To find Y-intercept, let x = 0 ● (6 , 0) Thus, (0 , 4) is a point on the line HOME

19 Writing Linear Equations
1. An equation of the line that passes through the point and has slope m is : Example : Find an equation of the line through (1 , 3) and its slope 2 Solution :

20 2. An equation of the line that passes through the point and is :
Example : Find an equation of the line through (-1 , 4) and (2 , -3) Solution : HOME

21 Parallel and Perpendicular Lines
Two lines are parallel if and only if their slope equal If slope k1 = m1 and slope k2 = m2 Parallel Two lines are Perpendicular if and only if the product of Their slope = -1 If slope h1 = m1 and slope h2 = m2 Perpendicular

22 Example : check the two lines parallel or perpendicular and
Solution : Let slope the first line is ,then Let slope second line is , then The line are not parallel The line are Perpendicular

23 Exercise, check the two line parallel or perpendicular
HOME

24 RELASI  Bola  Basket  Tari  Padus  I. Diagram panah
II. Pasangan berurutan {(Ali, Bola), (Bea ,Tari), (Cita, Basket),(Cita, Padus)} Siswa Ekskul  Bola  Basket  Tari  Padus Ali  Bea  Cita  III. Cartesius Siswa Ekskul    Ali Bea Cita Padus  Tari  Basket  Bola  

25 Produk Cartesius n(AXB)=6 A x B = B x A ?  n(A x B) = n(B x A)
Contoh : A = {a , b}  n (A) = 2 B = {1 , 2 , 3}  n(B) = 3 n(AXB)=6 A x B = B x A ? A x B = (a , 1) ….. B x A = (1 , a) …..  n(A x B) = n(B x A)

26 Pemetaan / Fungsi A ke B merupakan fungsi jika setiap anggota A mempunyai tepat 1 pasang anggota B A B A B A B A B a  b  c   1  2  3  4 a  b  c   1  2  3  4 a  b  c   1  2  3  4 a  b  c   1  2  3  4 BUKAN FUNGSI FUNGSI FUNGSI BUKAN FUNGSI O

27 A B Domain = daerah asal = {a , b, c } Kodomain = daerah hasil
= {1, 2, 3, 4} Range = hasil = {1, 2, 3}  1  2  3  4 a  b  c 

28 A B Nilai dari f(2) atau bayangan dari 2 atau peta dari 2  1 1   3
 5  7 1  2  3  Fungsi : Nilai dari f(2) atau bayangan dari 2 atau peta dari 2

29 Banyaknya pemetaan/fungsi
A = {a , b , c} B = {1, 2) A B A B A B A B a  b  c  1 2 a  b  c  1 2 a  b  c  1 2 a  b  c  1 2 A B A B A B A B a  b  c  1 2 a  b  c  1 2 a  b  c  1 2 a  b  c  1 2

30 Contoh: A = {a , b , c} B = {1, 2) n (A) = 3 n (B) = 2

31 n (A) = n(B) Korespondensi satu-satu A B A B A B A B
 A B Benar  A B Benar  A B Salah  A B Salah Definisi : fungsi yang memasangkan setiap anggota A (domain) tepat satu pada anggota B (kodomain) dan sebaliknya n (A) = n(B)

32 n (A) = n(B) = 3 maka banyaknya korespondensi satu – satu adalah ?
 A B  A B  A B  A B  A B  A B

33 n (A) = n(B) = a  banyaknya korespondensi satu –satu adalah a !

34 UH-2 Senin 21 November 2011 Materi: Relasi
Menyatakan relasi (diagram panah, pasangan berurutan, cartesius Produk cartesisu (AxB) Pemetaan/fungsi Banyak pemetaan/fungsi Nilai fungsi Korespondensi satu-satu