Use of symbols Objectives: F Grade Simplify expressions with one variable such as: a + 2a + 3a E Grade Simplify expressions with more than one variable such as: 2a + 5b + a – 2b D Grade Multiply out expressions with brackets such as: 3(x + 2) or 5(x - 2) Factorise expressions such as 6a + 8 and x2 – 3x C Grade Expand (and simplify) harder expressions with brackets such as: x(x2 - 5) and 3(x + 2) - 5(2x – 1)

Algebraic Definitions Expression An expression is a mathematical "phrase" that stands for a single number; for example, 3x + 1 Equation An expression that equals something, that maybe another expression or a single value. For example: 3x + 1 = 7 or 3x + 1 = 2x - 1 Variable A variable is a letter used in an algebraic expression in order to represent any number.

Algebraic Definitions Term A term is a number and / or variable(s) connected with x and / or ÷ separated from anther term by an ‘+’ or ‘-’ operation. For example 3x + 4y a b 3x + 4y term term term term

Use of symbols What combinations of letters and numbers mean a + a Can be read as 2 of those things called a So this can be written as 2 × a because things get abbreviated in maths we write this as: 2a a + a + a 3 of those things called a 3 × a So this becomes: 3a Also, because multiplication is commutable (the order of the Multipliers can be swapped and the answer remains the same) a× b = b × a or ab = ba

32 Use of symbols This is not to be confused with: a × a Can be read as a of those things called a So this can be written as a × a We know that 4 × 4 can be written as 42 because any number multiplied by itself like this Index number 32 a × a = a2 So the following are also true: Base number a × a × a = a3 The index number tells us how many times the base number is multiplied by itself. a × a × a × a = a4 e.g. 34 means 3 x 3 x 3 x 3 = 81

Use of symbols Collecting terms of add & subtract 2a + 3a a + a + a + a + a So this can be simplified to 5 of those things called a 5a 7a - 3a a + a + a + a + a + a + a - a - a - a So this can be simplified to 4 of those things called a 5a 7b - 3a b + b + b + b + b + b + v - a - a - a So this cannot be simplified because a and b are different This remains as: 7b - 3a

Use of symbols Now do these: 1. p + 2p + 3p 2. p + 4p − 3p 3. 2ab + 3ab 4. t + t + 4t 5. f + 6f − 10f 6. 5ad − 2da 7. d + 4d − 2d 8. h + h − 5h + 2h 9. p2 + p2 + p2 6p 2p 5ab 6t – 3f 5ad 3d – 1h 3p2

Use of symbols The index number of a letter or number only applies to the number or letter immediately preceding it. a3x = a × a × a × x abc3 = a × b × c × c × c ab3c = a × b × b × b × c Mathematical convention is that where we have more than one letter in a term, they are written in alphabetical order.

Use of symbols Further rules for the use of letters a + a = 2a a × a = a2 These are different types of terms and cannot be mixed If the index number in two terms is different they cannot be added If the index number is the same they can be added / subtracted Example: Simplify this expression 5x2 5x2 + 2x + 2x2 – 5x 2x + 2x2 -5x The same power of x these can be collected The same power of x these can be collected 7x2 - 3x

Use of symbols Now do these: 1. x + 3x + 5 + x 2. g + 2g + h 3. 7ab + 2b + 4a + 2ab 4. y + 2 + y + 4 5. 7w + 6 − 2w + 2 6. p2+p2+2p+p2+4p 7. d + 5 + 2d − 3 8. 2x + 3y + 4x + 2y 9. 4w + 2 + y − 3 5x + 5 3g + h 9ab + 4a + 2b 2y + 6 5w + 8 3p2+ 6p 3d + 2 6x + 5y 4w + y - 1 10. a2 + a3 + 2a2 11. 4a2b + 5a2b 12. 3cd2+4cd2−2dc2−3c2d 13. w5 + 2w5 + w 14. 5x3y + 2xy 3 + 2x 3 y 15. ct 2 + 3t 2 + t − t 2 16. 7abc 2 + 4ab 2 c + 5abc 2 + 3ba 2 c 3a2+ 6a3 9a2b 7cd 2- 5c 2 d 3w5+ w 7x3y + 2xy 3 2ct2+ 2t2 + t 12abc 2 + 4ab 2 c + 3ba 2 c

Use of symbols Summary F Grade Simplify expressions with one variable such as: a + 2a + 3a = 6a E Grade Simplify expressions with more than one variable such as: 2a + 5b + a – 2b = 3a + 3b Algebraic meanings: a + a + a = 3a a× b = b × a or ab = ba a × a × a = a3 If the index number in two terms is different they cannot be added If the index number is the same they can be added / subtracted The index number of a letter or number only applies to the number or letter immediately preceding it. Mathematical convention is that where we have more than one letter in a term, they are written in alphabetical order.