Unformatted text preview: MTH405 Wk 3 Lect 1 Algebra
Denitions
• Variable  the letters x, y and z which has the freedom to represent any number. • Coecient  A number multiplied to a variable.
• Term  a group of coecients 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. Coecient
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
• Dierence 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) (Dierence of Squares)
8. x2 − 16 = (x − 4)(x + 4) (Dierence of Squares) 3...
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 Spring '16
 Alveen Chand
 Algebra, Perfect square, following expressions, common factor

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