MTH405_Wk_3_Lect_1

# MTH405_Wk_3_Lect_1 - MTH405 Wk 3 Lect 1 Algebra Denitions...

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Unformatted text preview: MTH405 Wk 3 Lect 1 Algebra Denitions • Variable - the letters x, y and z which has the freedom to represent any number. • Coecient - A number multiplied to a variable. • Term - a group of coecients and variables multiplied or divided to each other. A term is separated from another term by + or − symbols. • Expression - A group of terms added or subtracted together. • Constant - A term which has only a number. • Equation - An expression which is equal to some value. Exercise. 5. Give examples of the following from the equation xy 2 +3y+5xy+23 = 1. Variable 2. Coecient 3. Term 4. Constant 5. Expression 1 Bracket Expansion Rule: When expanding brackets, the term outside is multiplied to every term inside the bracket. Exercise. Expand all the brackets of the following expressions. 1. (x + 2y) + (2x − y) 2. 2(x − y) − 3(y − x) 3. 2(p + 3q − r) − 4(r − q + 2p) + p 4. (a + b)(a + 2b) 5. (p + q)(3p − 2q) Evaluation of Expressions (Substitution) Exercise. Evaluate the following expressions. 1. Find the value of 2xy + 3yz − xyz , when x = 2, y = −2 and z = 4. 2. Evaluate 3pq 3 r3 when p = 23 , q = −2 and r = −1. 3. Find the sum of 3a, −2a, −6a, 5a and 4a. Simplifying Expressions Exercise. Simplify the following expressions. 1. Add together 2a + 3b + 4c, −5a − 2b + c and 4a − 5b − 6c. 2. Add together 3d + 4e, −2e + f , 2d − 3f , 4d − e + 2f − 3e. 3. From 4x − 3y + 2z , subtract x + 2y − 3z . 4. Subtract 23 a − 3b + c from b 2 − 4a − 3c. 5. Simplify (x2 y 3 z)(x3 yz 2 ) and evaluate when x = 12 , y = 2 and z = 3. 6. Simplify (a 2 bc−3 )(a 2 bc− 2 c) and evaluate when a = 3, b = 4 and c = 2. 3 7. Simplify a5 bc3 a2 b3 c2 1 1 and evaluate when a = 32 , b = 2 1 2 and c = 32 . Factorization There are 6 method of factorization. • Common factor • Factorization by Grouping • Type I factorization • Type II factorization • Dierence of Squares • Perfect Square Example. Factorize the following expressions. 1. 3a + 3b 2. 12b2 c3 − 8bc. 3. ax + bx + ay + by 4. x2 + 5x + 4 5. x2 + 7x + 12 6. 2x2 + 10x + 12 7. x2 − 9 8. x2 − 16 Solutions. 1. 3a + 3b = 3(a + b) (Common factor) 2. 12b2 c3 − 8bc = 4bc(3bc2 − 2). (Common factor) 3. ax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y). (Grouping) 4. x2 + 5x + 4 = (x + 4)(x + 1) (Type I) 5. x2 + 7x + 12 = (x + 3)(x + 4) (Type I) 6. 2x2 + 10x + 12 = 2x2 + 4x+ 6x+ 12 = 2x(x + 2) + 6(x + 2) = (x+ 2)(2x+ 6) (Type II Factorization) 7. x2 − 9 = (x + 3)(x − 3) (Dierence of Squares) 8. x2 − 16 = (x − 4)(x + 4) (Dierence of Squares) 3...
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