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- 1. [PART 2] By : Handayani Sinaga : 8146171034 Nadrah Afiati : 8146171056 Yulia Tiara Tanjung : 8146171090 Mathematic Education – Magister Program State University of Medan 2014
- 2. History of Algebra Algebra by Al-Khwarizmi Al-Khwarizmi's book on arithmetic synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero. The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. It is where we get the word algebra from: Restoration and equivalence Having introduced the natural numbers, al-
- 3. Algebra by Al- Khwarizmi He first reduces an equation (linear or quadratic) to one of six standard forms : Squares equal to roots. Squares equal to numbers. Roots equal to numbers. Squares and roots equal to numbers; e.g. x2 + 10 x = 39 Squares and numbers equal to roots; e.g. x2 + 21 = 10 x. Roots and numbers equal to squares; e.g. 3 x 2
- 4. Operation of Algebraic Expressions Addition and Subtraction Operation Multiplication and Division Operation Exponential Operation
- 5. Addition and Subtraction Operation In algebraic expressions, the terms that can be added and subtracted are only the similar terms or like terms. The like terms contain the same variable with the same exponent or terms containing constants. The procedures to run addition and subtraction of algebraic expression are as follows. Group the like terms Operate each group by adding or
- 6. Examples 1. Calculate (3x2 + 2x + 5) + (6x + 7) 2. Substract 2y2 + 11y from x2 + 8y2 + 7 Solutions : 1. (3x2+2x+5) + (6x+7) = 3x2 + (2x+6x) + (5+7) = 3x2 + 8x + 12 2. The sentence “Substract 2y2 + 11y from x2 + 8y2 + 7” means that you must run a substraction operation of (x2 + 8y2 + 7) – (2y2 + 11y). See that the like terms are 8y2 and 2y2. Therefore, (x2 + 8y2 + 7) – (2y2 + 11y) = x2 + 8y2 + 7 – 2y2 - 11y = x2 + (8y2 -2y2) - 11y + 7
- 7. Multiplication and Division Operation Multiplications of one terms and two terms fo algebraic expressions. These multiplications can be done using distributive characteristic below. a(x ± y) = ax ± ay or (x ± y)a = ax ± ay Multiplications of two terms and two terms fo algebraic expressions. These multiplications can be done using distributive characteristic or using multiplication scheme. (x + y) (a + b) = x(a + b) + y (a + b) or (x + y) (a + b) = ax + bx + ay + by = ax + bx + ay + by
- 8. Examples Expand the following multiplication of algebraic expressions! 1. 2(x + 3) 2. 3(x + 5) + 5(x - 3) Solution : 1. 2(x + 3) = 2x + 2 (3) = 2x + 6 2. 3(x + 5) + 5(x - 3)= 3x + 3(5) + 5x + 5(3) = 3x + 15 + 5x – 15 = (3 + 5)x + 15 – 15 = 8x
- 9. Division In the division of algebraic expressions is known 2 key terms, i.e : - Divisions with similar terms, such as 2x : x, and - Divisions with different terms, such as x2 : x See the following example to know how to conduct a division of algebraic expressions : We can use the properties of exponential within integer to solve division of algebraic expressions. 𝑥2 + 2𝑥 : 𝑥 = 𝑥2 + 2𝑥 𝑥 = 𝑥2 𝑥 + 2𝑥 2 = 𝑥 + 2
- 10. Exponential Operation Exponential is a recur multiplication of number. We can expand pn as follow : pn = p x x p x ... x p n terms Exponential properties of integers may apply to algebraic forms. Exponential of algebraic forms shows the multiplication of an algebraic form to it self. Examples : Determine theexponential result of the algebraic expressions below. 1. (2x + 4y)2 = (2x)2 + 2(2x)(4y) + (4y)2 = 4x2 + 16xy + 16y2 2. (x – y – z)2 = {(x – y) – z}2 = (x – y)2 + 2(x - y)(-z) + z2
- 11. Fractions Operation of Algebraic Expressions Simplify Algebraic Expressions in Fraction Addition and Substraction Operation in Fraction Multiplication and Division Operation in Fraction
- 12. Simplify Algebraic Expressions in Fraction The fraction in algebraic expressions can be simplified by dividing the nominator and the denominator in the fraction by the common factor of the nominator and the denominator. Example : Simplify this fraction : 6𝑥2+12𝑥 9𝑥 Solution : 6𝑥2 + 12𝑥 9𝑥 = 3𝑥 (2𝑥 + 4) 3𝑥(3) = 2𝑥 + 4 3
- 13. Addition and Substraction Operation in Fraction The algebraic fractions can be added or subtracted if the fractions have the same denominator. However, if the fractions that will be added or subtracted have different denominators then denominators must be equalized first. We can equalize the different fractions denominators by the following ways. Find the Least Common Multiple (LCM) of the denominators Perform the given fractions into equivalent
- 14. Example Perform the addition and subtraction for the following fractions. 1. 𝑥 3𝑦 + 4𝑥−𝑦 3𝑦 2. 6𝑚 𝑚−1 − 4 𝑚2−1 Solutions : 1. 𝑥 3𝑦 + 4𝑥−𝑦 3𝑦 = 𝑥+4𝑥−𝑦 3𝑦 = 5𝑥−𝑦 3𝑦 2. The denominators of 6𝑚 𝑚−1 and 4 𝑚2−1 are m-1 and m2 -1 respectively. The LCM of m-1 and m2 -1 is m2 -1 = (m-1)(m+1), thus 6𝑚 𝑚−1 − 4 𝑚2−1 = 6𝑚(𝑚+1) 𝑚−1 (𝑚+1) − 4 𝑚−1 (𝑚+1) = 6𝑚 𝑚+1 − 4 𝑚−1 (𝑚+1) = 6𝑚2+6𝑚−4 𝑚−1 (𝑚+1)
- 15. Multiplication and Division Operation in Fraction Multiplication with algebraic fractions can be done by multiplying both the nominators and multiplying both the denominators. Then, the result is commonly simplified. To divide fractions, say 𝑥 𝑦 with 𝑎 𝑏 , can be done by multiplying fractions 𝑥 𝑦 with 𝑏 𝑎 . Therefore, 𝑥 𝑦 ∶ 𝑎 𝑏 = 𝑥 𝑦 𝑥 𝑏 𝑎 = 𝑥𝑏 𝑦𝑎
- 16. Examples Perform the multiplication and division of fractions below 1. 7𝑦 2𝑥 𝑥 𝑥𝑦 14𝑦 2. 𝑚 𝑚+1 : 𝑚+3 𝑚+1 Solutions : 1. 7𝑦 2𝑥 𝑥 𝑥𝑦 14𝑦 = 7𝑥𝑦2 28𝑥𝑦 = 𝑦 4 3. 𝑚 𝑚+1 : 𝑚+3 𝑚+1 = 𝑚 𝑚+1 𝑥 𝑚+1 𝑚+3 = 𝑚 𝑚+3
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