E xpansions expand and simplify

w hy A lgebra ? Why Algebra ? Imagine that mum promises you €2 everyday for the following week. Then you can calculate the “total amount” of euro you will have from Monday to Sunday of next week, i.e. €2 multiplied by 7 days = €14. Now imagine mum promises you €2 for each day upon which your behaviour is good in the following week. ☺ You cannot calculate the “total amount” at the beginning of the week as you do not know the days when your behaviour is good beforehand. The number of days when you behave can very from “0” to “7” and hence the “total amount” can vary from “€0” to “€14”. Therefore Algebra is a tool by which we can “generalise” situations to which we don’t have an “immediate” answer. T = 2D is an Algebraic Equation which would give the “total amount” T for the number of days D upon which you obeyed. “D” and “T” are called variables. D can take any of the values 0, 1, 2, 3, 4, 5, 6, 7 and T varies accordingly 0, 2, 4, 6, 8, 10, 12, 14. The number 2 in the equation is said to be a constant as mum will not change the amount of euro that she has agreed upon per day. So if you have behaved for 3 days the amount you will receive is T=2D T=2(3) T= €6 in total

s ome g roundwork What is a an algebraic term ? An algebraic term is a number (constant), a letter/s (variable/s) or a multiplication/division of both. e.g. 5 is a constant (maintains the same value). x, y, z are variables (can represent any value). 5xyz is a multiplication of constants and variables resulting in a variable itself. The above examples 5, x, y, z and 5xyz are all terms and they can all be called “monomials”. So this leads us to answer the question : What is a mathematical expression? A mathematical expression is a number of terms “concatenated” using any of the Four (white lie) Basic Operations PLUS / SUBTRACT / MULTIPLY / DIVIDE. “+” “-” “x” “÷” possibly more than once. e.g x – 7xyz is an expression consisting of three terms: a “trinomial” 3x 2 + 8x is an expression consisting of two terms: a “binomial” 5ab + 2b – 4 3 c

…….. g roundwork….. f undamental r ule + & - What are “like terms”? 7 and 3 are like terms, they are both Integers and can be “collected” under normal addition or subtraction: e.g = 10 ; 7 – 3 = 4 ; 3 – 7 = - 4. “5a” and “3a” are also like terms and can be collected under addition and subtraction : e.g. 5a + 3a = 8a ; 3a – 5a = -2a Similarly 4xyz and -7xyz e.g. 4xyz + ( -7xyz ) = -3xyz and 4xyz – (-7xyz ) = 11xyz. Unless terms are “like terms” in Algebra they cannot be collected and hence they cannot be added or subtracted. ‘ to operate on algebraic terms using the Binary operations “+” and “ –” the terms must be “like terms” a and a 2 are not like terms. For like terms not only must the letter/s be the same but also the power / index. 5a 3 b 2 and 7a 3 b 2 are like terms

…….. g roundwork….. f undamental r ule X & ÷ To operate on algebraic terms using the operations “x” and “÷” the terms do not have to be like terms. If terms are not like join the letters or write over each other depending on whether you are multiplying or dividing, otherwise if the terms are like or are a different power to the same letter ( the base is the same) use the rules to indices. e.g. 1 ( x ) ( x ) = x 2. e.g. 2 ( y 3 ) ( x 4 ) = y 3 x 4. e.g. 3 (5xy)(3xz) = 15x 2 yz e.g. 4 ( 3x ) ÷ ( 5y ) = 3x 5y e.g. 5 ( 4x 2 )(3x 3 ) = 12x 5. e.g. 6 ( 5xy 2 )( 3x 4 y 3 ) = 15x 6 y 5. e.g. 7 ( 2x 5 y 4 z 3 ) ÷ ( 4x 2 yz 2 ) = 1x 3 y 3 z 2

e xpand & s implify……some useful points Expand brackets and collect like terms. Be careful with minus signs before brackets. Constants written before the brackets should be expanded last especially if there is a minus before the brackets. When multiplying or dividing algebraic terms,the rules to Indices are applied whereas when we add or subtract terms we say we are “collecting like terms” A power is not distributed over a ” + “or a “ – “ but over a “x” and a “/ ”.

e xpand a nd s implify…..some simple examples e1 Expand and Simplify 3 ( 2x + 1 ) = 3 ( 2x ) + 3 ( 1 ) = 6x + 3. Ans. e2 Expand and Simplify 3 ( 2x + 1 ) + 4 ( 3x - 7 ) = 3 ( 2x ) + 3 ( 1 ) + 4 (3x ) – 4 ( 7 ) = 6x x – 28 = 6x + 12x + 3 – 28. = 18x – 25. e3 Expand and Simplify 4 ( 2x – 5y ) = 4 ( 2x ) - 4 ( 5y ) = 8x – 20y. Ans. e4 Expand and Simplify 7 ( 3x - y ) - 9 ( 3x – 4y ) = 7 ( 3x ) - 7 ( y ) - 9 (3x ) – 9 ( 4y ) = 21x – 7y - 27x – 36y = 21x - 27x – 7y – 36y. = -6x – 43y.

e6 Expand and Simplify (a + b) 2 = ( a + b )( a + b ) = a (a) + a (b) + b (a) + b (b) = a 2 + ab + ba + b 2 = a 2 + 2ab + b 2. Note that a power cannot be distributed over a “+” or a “-” e xpand a nd s implify………. e5 Expand and Simplify -3 ( 4x - 9 ) = -3 ( 4x ) + 3 ( 9 ) = -12x Ans. e6 Expand and Simplify -3 ( 2x + y ) - 4 ( 3x – 8y ) = -3( 2x ) - 3 ( y ) - 4 (3x ) + 4 ( 8y ) = -6x – 3y - 12x + 32y. = -6x - 12x – 3y + 32y. = -18x + 29y.

e xpand a nd s implify…..some harder examples Constants written before the brackets should be expanded last especially if there is a minus before the brackets. When a bracket has a index of “2” we call it a perfect square. A quick way to expand a perfect square is the following : Expand (3a - 2b) 2 Square the first term :(3a) 2 = 9a 2 Square the last term :(-2b) 2 = 4b 2 Multiply the terms and the result by 2 : (3a)(-2b) = (-6ab) ; 2(-6ab) = -12ab. Solution 9a ab + 4b 2. e6 Expand and Simplify (3a - 2b) 2 = ( 3a - 2b )( 3a -2b ) = 3a (3a) - 3a (2b) - 2b (3a) + 2b (2b) = 9a 2 - 6ab - 6ba + 4b 2 = 9a ab + 4b 2. e7 Expand and Simplify -3 ( 4x - 9 ) 2 = -3 (4x – 9 )(4x – 9 ) = -3 ((4x)(4x) – (4x) (9) – (9)(4x) + (9)(9)) = -3 (16x 2 – 36x – 36x + 81 ) = -3 (16x 2 – 72x + 81) = - 48x x – 243

e xpand a nd s implify…..some more harder examples e8 Expand and Simplify ( 4x - 9 )(5)(7x + 3) Putting the constant at the very front we get = 5 ( 4x – 9 )( 7x – 9 ) = 5 ((4x)(7x) – (4x)(9) – (9)(7x) + (9)(9)) = 5 (28x 2 – 36x – 63x + 81) = 5 (28x 2 – 99x + 81) = 140x 2 – 495x Ans e9 Expand and Simplify ( 4x - 9 )(5)(7x + 3)(-3) Putting the constants at the very front we get = (-3)(5) ( 4x – 9 )( 7x – 9 ) = -15 ( 4x – 9 )( 7x – 9 ) = -15 ((4x)(7x) – (4x)(9) – (9)(7x) + (9)(9)) = -15 (28x 2 – 36x – 63x + 81) = -15 (28x 2 – 99x + 81) = - 420x x Ans