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Academic Skills Advice Algebra Refresher Sheet 1 Writing Algebraic Expressions: Writing an algebraic expression is like writing a sentence in maths instead of English. You do this by assigning letters to numbers. An algebraic expression is a set of instructions on how to perform a calculation. Examples: Write the following as an algebraic expression: Five times a number minus three times another number. First I need to assign letters to the ‘unknown’ numbers. I will call the first one ‘n’ and the second one ‘m’ so now I have: Five times n minus three times m. (Notice I have replaced the 1st and 2nd number with ‘n’ and ‘m’). Next I replace the words with maths symbols so that I have: 5xn–3xm Tidy up: 5n – 3m This is our expression. A number divided by three: can be written as Half of a number plus quarter of another number: can be written + You can choose any letter but make sure that if you want it to mean a different number you choose a different letter. Simplifying Expressions: Once you have an algebraic expression it can be simplified by collecting all the ‘like terms’ together (i.e. combining things that are the same letter or combination of letters). If an expression includes brackets then you may need to multiply out the brackets first to see what will combine. Examples: Simplify the following expression: 3a + 7b – 2c – 4b – 6c + a Collect together any letters that are the same: 3a + a = 4a 7b – 4b = 3b -2c – 6c = -8c So we have: 4a + 3b – 8c © H Jackson 2008 / Academic Skills 1 Simplify the following: 3a (a – b) + 5ab First remove the bracket to see what you can combine. To do this you need to multiply the bit outside the bracket by every bit (term) inside the bracket. 3a x a = 3a2 3a x –b = - 3ab So after multiplying out the brackets we have: 3a2 - 3ab + 5ab = 3a2 + 2ab Factorisation: An expression can be simplified further by factorising it. This means taking out any common factors (i.e. numbers or letters that are in every term of the expression). This involves putting the brackets back in. Examples: Simplify the following: 8 + 3 + 9 − 4 Collect like terms: 8 − 4 = 4 3 + 9 = 12 4 + 12 So we have: Now we can factorise because each term will divide by 4 and . (This is the opposite of multiplying out the bracket.) 4 4 12 =1 4 = 3 Put the bit you have divided by outside the bracket. 4(1 + 3) So now we have: Simplify the following expression: 3ab2 + 4a2b – b2a – 5ab – 6a2b This looks more complicated but remember you can collect any terms that are the same. Notice that ab2 is the same as b2a (just written the other way round) but is not the same as a2b. Collect like terms: 3ab2 – b2a = 2ab2 4a2b – 6a2b = -2a2b -5ab (there is no other ab term) So we have: 2ab2 - 2a2b - 5ab Now we have an expression that we can factorise as each term includes an ‘a’ and a ‘b’ so we can divide each term by ab. 2 2 −22 = 2 So we have = −2 ab(2b -2a -5) © H Jackson 2008 / Academic Skills −5 = −5 (this looks tidier and simpler) 2 Practice Questions: Write algebraic expressions for the following: 1) 2) 3) 4) 5) Four times a number plus seven. Six times a number take away five times another number. Three times a number plus a number divided by four take away seven. A number plus two all multiplied by three. Three times a number plus four all divided by three. Simplify the following: 1) 3d + 7d 2) 8e – 5e 3) 3d + 4e – 2d + 6e 4) d2 + 3d – e2 + 4d 5) -4 × 3d 6) 12d2ef3 4de2f 7) 9e2d 27e2d2 8) 8( + 2) + 3(2 − 3) 9) 3(2 + 2) + 4( − 2) − 2( − 3) 10) 15 + 4 − 5 + 6 11) 6a + 6b 12) 3b – 3b2 13) 4 2 + 16 14) 27c3 – 9c 15) 12mn – 4nm +12m © H Jackson 2008 / Academic Skills 3