Topics Covered: Rules of indices Expanding bracketed expressions Factorisation of simple algebraic expressions

Transcription

1 Algebr Workshop: Algebr Topics Covered: Rules of indices Expnding brcketed expressions Fctoristion of siple lgebric expressions Powers A power is used when nuber is ultiplied by itself severl ties. For exple: power bse is interpreted s bse to the power of, i.e. ( 8). So is being ultiplied by itself three ties. The ter power is ore often referred to s index. (NB: Plurl of index is indices) Key Points: Any nuber (or bse) rised to the power of is itself, e.g.. More generlly,. Any nuber rised to the power of 0 is lwys, e.g. 0. More generlly, 0. Resoning: 0 However, dividing nuber by itself gives, therefore, 0.

2 Algebr The rules of indices The rules of indices re used to nipulte lgebric nd nuericl expressions. These rules cn only be pplied to ters tht hve the se bse.. n +n Exples: 4 6 x x 0 x xy x y 4 6x y. n n n Exples: x y xy 9 xy 4 n. ( ) n n Exples: (6 ) 6 This rule (rule ), cn lso be used to nipulte expressions such s: ( b n ) k k b nk

3 Algebr For exple, ( b ) 4 b 8 (x y) 4 6x 0 y 4 Negtive Powers: Theory: Let s look t + n n n rule, this cn be re-written s: -(+n) -n., clerly this is the se s n, using Therefore, when nuber is rised to negtive power, e.g. -, it cn be rewritten s:, i.e. - / Exple: 4-4 Roots: A bsic root tht you should know is In generl, x x. Positive integer x n n x (where n is nturl nuber) Tking this further, x n n ( x ) ( x ) n. Exples: ( ) ( ) ( ) 9 7 7

4 Algebr Questions (Rules of Indices): Siplify the following: x 6 x 9. ( bc )(b c) 4. 6x y. xy 6. ( - ) 7. (x y ) 4 8. (x - y ) (Solutions on pge 8) Expnding Brcketed Expressions When brckets re reoved (or expnded) fro round n lgebric expression, ny fctors outside the brckets ust be ultiplied to ech ter inside the brckets. For exple, (x + y) is the fctor nd is directly outside the brckets. Therefore, when expnding (or reoving) the brckets, the fctor hs to be ultiplied to both x nd y inside the brckets, i.e. (x + y) x + y In siilr wy, if the fctor outside the brckets is negtive nuber/expression, the negtive fctor hs to be ultiplied to every ter within the brckets, i.e. -(x + y) -x y In the se wy, (x - y) x y -(x - y) -x + y Exples:. b( + bc) 6 b + b c. -x y(xy 4x + ) -0x y + 0x y x y. xy z(x + 4xz y z + 6) 6x y z + x y z xy z + 8xy z 4

5 Algebr Questions (Expnding brcketed expressions): Reove the brckets fro the following brcketed expressions:. x (4 x). -4( b + 4c). 6(y + ) y 4. (x y) + (x 6y) (Solutions on pge 8) Multiplying together two brcketed ters Method : Siley fce ( + b)(c + d) Method : FOIL (First, Outer, Inner, Lst) F(irst) The product of the two first ters in ech brcket O(uter) The product of the two outer ters I(nner) The product of the two inner ters L(st) The product of the two lst ters in ech brcket F O ( + b)(c + d) I L Soe generic expnsions: ( + b) ( + b)( + b) + b + b + b + b + b ( + b)( - b) b + b b b ( b)( + b) + b b b b ( b)( b) b b + b b + b Difference of squres

6 Algebr Exples:. (x + )(x ) x x + x 0 x + x 0. (x )(y + ) xy + x y. (x 6)(x ) x 4x 6x + x 0x + 4. (4x + )(x + y) 8x + xy + 0x + y Questions (Multiplying out two brckets): Expnd the following:. (x 4)(x + ). (x + 7)(x ). (x )(x ) 4. (x y)(4x + y). (x + )(x x + 4) (Solutions on pge 8) Fctoristion of Algebric Expressions Fctoristion is the reversl of expnding brckets. Coon fctors re identified nd tken outside brcketed ter. You cn lwys check whether your fctorised expression is correct by expnding/reoving the brckets nd checking you get the originl expression. (b + c) b + c Exples: (b c) b c. 4x + 4(x + ) -(b + c) -b c. 8x -(b c) -b + c 6x 8x(x ) ( + b)( + b) + b + b. 0x + x x (x + ) ( - b)( - b) - b + b 4. 9x x + 6xy x(x + y) ( + b)( - b) - b 6

7 Algebr Questions (Fctoristion of lgebric expressions): Fctorise the following lgebric expressions. 7x +. 0y. 9x + x 4. 4x 0xy. x + 0x 6. 6y 4 6y + 8y 7. 8x 4 y + 0x y 6x y 8. x y + 6x (Solutions on pge 8) 7

8 Algebr Solutions (Rules of Indices):. 4. x. b c 4.. xy x 8 y 8. x -6 y ( ). 6 ( 4) 64 ( ) ( ) Solutions (Expnding brcketed expressions):. x (4 x) 0x x. -4( b + 4c) b 6 c. 6(y + ) y 6y + y y + 4. (x y) + (x 6y) x y + 4x y 9x y Solutions (Multiplying out two brckets):. (x 4)(x + ) x 4x + x 0 x + x 0. (x + 7)(x ) x 4x + 7x 4 x + x 4. (x )(x ) x x 0x + x x + 4. (x y)(4x + y) x + xy 8xy 0y x + 7xy 0y. (x + )(x x + 4) x x 4 + 4x + x x + 4 -x 4 4x + x + 4x + 4 Solutions (Fctoristion of lgebric expressions):. 7(x + ). ( y). x(x + ) 4. 4x(x y). x (x + ) 6. 6y(6y + y) 7. x y(4x + y y ) 8. 6x (y + x ) 8