Introduction to Algebra Unknown values can be represented by letters or symbols (called variables) Rules can be represented in a general form Words can be translated into mathematical expressions

Variable Expressions 4p means 4 times p 5xy means 5 times x times y A number written immediately beside a variable implies multiplication 4p means 4 times p 5xy means 5 times x times y -3m2 means -3 times m times m

The Replacement Property A variable in an expression can be evaluated by substituting its replacement value Example: evaluate 4p for p = 7 4p = 4(7) <===Note <===Note <===Note Always put the replacement value in a bracket = 28

5xy 5(3)y 5(3)(-2) =-30 -3m2 -3(4)2 =-3(16) =-48 Evaluate 5xy for x = 3, y= -2 5xy 5(3)y 5(3)(-2) =-30 Evaluate -3m2 for m = 4 -3m2 -3(4)2 =-3(16) =-48

Terms and Polynomials 2xy 7 –3x2 m -y2 6xy –3 x2 xy 2x2 A term is defined as the product of variables and constants. 2xy 7 –3x2 m -y2 Like terms have identical variable parts 6xy –3 x2 xy 2x2

Simplifying Polynomial Expressions Expressions can be simplified by combining like terms. Remember: everything’s an addition, but some of the terms are negative. Add the terms using Integer rules. The sign on the left of each term moves with the term Re-order the terms Simplify the following: 1. 3x – 5y + 2 – 7y + 2 – x = 3x – x – 5y – 7y + 2 + 2 Add the coefficients of the like terms = 2x – 12y + 4 Re-order the terms 2. 2x2 – 3 + 5x – 9x + 3 + 7x2 = 2x2 + 7x2 + 5x – 9x + 3 – 3 = 9x2 – 4x Add the coefficients of the like terms

Adding Polynomials Polynomials are groups of terms. To add them, brackets have to be removed (associative property), and then the terms can be re-ordered (commutative property). Add the polynomials: 1. (x + 5y – 2) + (7y – 3x + 7) = x + 5y – 2 + 7y – 3x + 7 Nothing changes when removing the brackets = x – 3x + 5y + 7y – 2 + 7 Re-order, and simplify like terms Re-order, and simplify like terms Change double signs to a single sign = -2x + 12y + 5 2. (-x2 + x) + (2x – 5x2) + (-8 + x2) = -x2 + x + 2x – 5x2 + -8 + x2 = -x2 + x + 2x – 5x2 – 8 + x2 Nothing changes when removing the brackets = -x2 – 5x2 + x2 + x + 2x – 8 = -5x2 + 3x – 8

Expanding brackets When a number is multiplied by a bracket, use the distributive property to expand the bracket. Think of it as: 2(x + -3) Multiply the 2 by each of the terms in the bracket Expand the following: 1. 2(x – 3) Simplify = 2(x) + 2(-3) Usually you won’t even show this step Multiply the 5 by each of the terms in the bracket = 2x – 6 Think of it as: 5(x + -xy + 4) 2. 5(x – xy + 4) Simplify Usually you won’t even show this step = 5(x) + 5(-xy) + 5(4) = 5x – 5xy + 20

Watch out for negatives! If there’s a negative in front of the bracket, it gets multiplied by each of the terms in the bracket. Think of it as: -1(2a + -5) Expand the following: Multiply the negative by each of the terms in the bracket 1. -(2a – 5) Simplify = -(2a) + -(-5) Usually you won’t even show this step Multiply the -3 by each of the terms in the bracket = -2a + 5 Think of it as: -3(w + -2 + d) 2. -3(w - 2 + d) Simplify = -3(w) + -3(-2) + -3(d) Usually you won’t even show this step = -3w + 6 - 3d

Putting it all together! Expand the brackets, and then collect the like terms Don’t forget to multiply in the negative Simplify the following: 1. 5(x - 4) – 2(-3 + x) = 5x – 20 + 6 - 2x = 3x - 14 Don’t forget to multiply in the negative 2. -x(y + 7) + 4(x – 3 + xy) = -xy – 7x + 4x – 12 + 4xy = 3xy - 3x - 12