Revision Topic 14: Algebra

Transcription

1 Revision Topi 1: Algebr Indies: At Grde B nd C levels, you should be fmilir with the following rules of indies: b b y y y i.e. dd powers when multiplying; y b b y y i.e. subtrt powers when dividing; b b ( y ) y i.e. when you hve power of power, multiply powers together. y 0 1 i.e. nything to the power 0 is 1. Exmple: Simplify eh of the following, if possible: ) ) 1 9 ) 0 ( x ) d) (dd powers together). simplifying top subtrting powers e e) b 0 ) (x) 1 (nything to the power 0 is 1) 6 d) e e (multiply powers together) e) b this nnot be simplified s the bse numbers ( nd re different letters. Exmple : Work out the vlue of eh of the following: ) ) ) (nything to the power 0 is 1) ) Exmple : Simplify: 7 1y ) x x ) 6 b b y ) x x x x 1x (i.e. multiply together the numbers nd dd the powers) ) 1y y 7 1 y 7 y (i.e. divide numbers nd subtrt powers) 6 b b (6 ) ( ) ( b b ) b. 6 Dr Dunombe Ester 00 1

2 Exmintion Style Question Find the vlue of x in eh of the following: 6 x 6 x ) ) (7 6 ) x 7 d) 7 x 0 Exmintion Style Question : Simplify fully eh of these expressions. Leve your nswers in power form. ) ) 6 d) 9 9 e) 6. Exmintion Style Question : Simplify eh of the following expressions. ) 8 6x x ) x y x y Dr Dunombe Ester 00

3 Expnding brkets Expnding out single brket: You n remove single brket by multiplying everything inside the brket by the number, or expression, on the outside. Exmple Expnd the following brkets: ) 6(7d ) y(8y x + 1) ) x (x x ) d) xy(x y) ) 6(7d ): Multiply both the 7d nd the by the number on the outside: We get 6(7d ) = d - y(8y x + 1): Multiply everything in the brket by y: We get: y(8y x + 1) = 8y xy + y ) x (x x ) : Multiply ll the terms inside the brket by x: We get x(x x ) 10x 1x 10x d) xy(x y): Multiply the x nd y by xy: This gives: xy(x y) = 8x y xy. You need to tke re when there is minus sign in front of brket. Exmple : Expnd nd simplify: ) -(x y) 6x(x + ) (x ) ) Here we multiply everything in the brket by -. This gives: -(x y) = -8x + 1y If we multiply out the first brket we get: 6x(x + ) = 18x + 1x If we multiply out the seond brket, we get: (x ) = -1x + 6. Putting it ll together: 6x(x + ) (x ) = 18x + 1x - 1x + 6 = 18x x y = +1y s two minuses multiply to mke plus! Exmintion Question: ) Multiply out: t ( t t ). Multiply out nd simplify: ( + 6) ( 6) 1 b ) Simplify:. b Dr Dunombe Ester 00

4 Expnding out double brkets: When there is pir of brkets multiplied together, you need to multiply everything in the first brket by everything in the seond. Exmple: Multiply out the following brkets: ) (x )(x + ) (x y)(x ) ) (x + y) ) (x )(x + ) We n expnd these brkets diretly, multiplying everything in the first brket by the terms in the seond brket. This gives: (x )(x + ) = x + 1x x 8 = x + 10x 8. Alterntively, you n drw grid to help expnd the brkets: x - x x -x 1x -8 Adding the numbers inside the grid gives: (x )(x + ) = x + 1x x 8 = x + 10x 8. (x y)(x ) The grid for this would look like: x -y x x -6xy - -8x +1y Adding the numbers inside the grid gives: (x y)(x ) = x 6xy 8x + 1y This nnot be simplified! ) (x + y) To squre brket, you multiply it by itself! Drwing grid: x y x x 6xy y 6xy 9y Adding the expressions inside the grid: (x + y)(x + y) = x + 6xy + 6xy + 9y = x + 1xy + 9y Exmintion Questions: Expnd nd simplify: 1) (x y)(x + y) ) (x )(x + ) ) (x y) Dr Dunombe Ester 00

5 Ftorising Ftorising is the reverse of multiplying out brkets, i.e. when you ftorise n expression you need to put brkets bk into n expression. Common ftors: Some expressions n be ftorised by finding ommon ftors. Exmple: Ftorise the following expressions. ) 1e + 18 xy + x ) x 6x d) x + x e) 10x y 1xy. ) We look for ommon ftor of 1e nd 18. We notie tht 6 goes into both of them. We therefore write 6 outside brket: 1e + 18 = 6(e + ) 6 goes into 1 twie 6 goes into 18 times We notie tht x ppers in both xy nd x. This n be tken outside brket: xy + x = x y + x = x( y + ) = x(y + ) ) As x is x x, both x nd 6x hve x s ftor. So, x 6x = x x 6x = x(x 6) d) Looking t the number prts, we notie tht is ommon ftor of both nd. x + x = (x + x). This hsn t been ompletely ftorised yet, s both x nd x lso ontin n x. We therefore n n x outside the brket. x + x = (x + x) = x(x + 1). x is 1x, so when x is tken outside the brket, we re left with 1 inside. e) 10x y 1xy : Looking t the numbers, we see tht both 10 nd 1 hve s ftor. Both terms lso hve n x nd y in ommon. We n therefore ftorise by writing xy in front of brket. 10x y 1xy = x x y x y y = xy(x y) Note: You n hek your nswers by expnding out the brkets. Exmintion Question Ftorise ompletely: () x x ( p q + pq. Dr Dunombe Ester 00

6 Exmintion Question : ) Expnd nd simplify: (x + 1) (x ). Expnd nd simplify: (x + )(x ). ) Ftorise ompletely: 6 9b. Ftorising qudrtis Simple qudrtis like x x or x 7x 1 n often be ftorised into two brkets. Generl steps for ftorising x bx Step 1: Find two numbers tht multiply to mke nd dd to mke b. Step : Write these two numbers in the brkets: (x )(x ) Exmple: Ftorise x 9x 18 Step 1: Find two numbers tht multiply to mke 18 nd dd to give 9. These numbers re 6 nd. Step : The ftorised expression is (x + 9)(x + ) Exmple : Ftorise x 7x 1 We need to find two numbers tht multiply to mke 1 nd dd to give -7. These numbers re - nd -. So the nswer is (x )(x ). Exmple : ) Ftorise x 8x 0 Solve x 8x 0 0 ) We hve to find two numbers tht multiply to mke -0 nd dd to give -8. These re -10 nd. The ftorised expressions is (x 10)(x + ). To solve the eqution x 8x 0 0 we use our ftorised expression: (x 10)(x + ) = 0. We hve two brkets tht multiply together to mke 0. The only wy this n hppen is if one of the brkets is 0. If the first brket is 0, then x 10 = 0, i.e. x = 10. If the seond brket is 0, then x + = 0, i.e. x = -. So the solutions re x = 10 nd x = -. Exmintion question Ftorise x x 1. Hene or otherwise solve x x 1 0. Dr Dunombe Ester 00 6

7 Exmintion question: ) Ftorise x + 8y. Ftorise ompletely 6. ) Ftorise x 9x 18. Exmintion question: ) Expnd nd simplify (x + )(x ). Ftorise x x 1. ) Solve x x 1= 0. Differene of two squres When you expnd out the brkets for (x + )(x ) you get x + x x whih simplifies to x. The result x = (x + )(x ) is lled the differene of two squres result. Exmples: 1) y 16 = y = (y + )(y ). ) z = z = (z + )(z ). ) 9x y = (x) (y) = (x + y)(x y). Exmintion question: ) Ftorise x y. Use your nswer to ) to work out the EXACT nswer to Dr Dunombe Ester 00 7