Simplifying Expressions: Algebraic Properties |
Much of our work in mathematics consists of simplifying mathematical expressions so that we can better understand what they are, what they mean and what they represent. Say we're working on a word problem in which we use x to represent the unknown or the value we're looking for. We formulate our algebraic statement of the problem and it is:
4(3x – 5) – 7(2x – 3) = 32.
It would be hard to come to any conclusions about the value of x from this mess, but if we apply the distributive property and collect like terms, it becomes simpler.
We get: 12x – 20 – 14x + 21 = 32
Now we reorder the terms and get: 12x – 14x – 20 + 21 = 32
when we collect like terms we get – 2x + 1 = 32
therefore, – 2x = 31 which means x = – 15.5
Notice that as always in math, the words tell us precisely what to do. The distributive property tells us to distribute the multiplication by 4 and by – 7 on the contents of the brackets that follow them. Once we've done this, we collect like terms -- or apply the associative property. This property tells us we can add a list of values in any order we want and we'll get the same sum as long as we include all the items in the list.
It says the sum 9 + 3 + 7 is exactly equal to the sum 7 + 9 + 3 or 3 + 9 + 7 etc.
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Once we know the properties, all we need to know is in what order to apply them.
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The properties we discuss here are really the rules of behaviour for Real numbers and the algebraic variables we use to represent Real number values. They describe how the various elements of algebra behave in relation to each other and to their position and order in the expressions and equations we simplify and solve.
1. The Distributive Property
This property tells us how to "remove brackets" -- or "multiply out the brackets".
To distribute is to hand out -- to "spread" out -- and that's exactly what this property says.
If a, x and y are any three Real numbers, then: a(x + y) = ax + ay |
And since the commutative property says that the order doesn't matter when we multiply and add, we can say:
(x + y)a = ax + ay
Notice how we distribute the multiplication by "a" on the contents of the bracket that precedes or follows it. Notice also how we write the variables in alphabetical order.
Even though xa + ya is exactly the same as our answer, we must respect mathematical convention so we write the variables in alphabetical order whenever possible. Another convention is to write expressions in ascending or descending powers of the variable.
So 3x^{2} – 5x + 57 is more acceptable than – 5x + 57 + 3x^{2} or – 5x + 3x² + 57
A common mistake for beginners is to ignore the sign of the multiplier. For example:
– 3(b – c – 5d) = – 3b + 3c + 15d.
Since we're multiplying by NEGATIVE 3, we get POSITIVES
from the – c and – 5d in the bracket.
Examples
y(a + b – 3c) = ay + by – 3cy | – 4y(2a – 5b – c) = – 8ay + 20by + 4cy |
(a + b – 3c) 9y = 9ay + 9by – 27cy | –3a(a + b – 3c) = –3a² – 3ab + 9ac |
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2. The Associative Property
This property says that we can add a list of values in any order or "grouping", as long as we "associate" or "group" them all. In other words (7 + 9) + 3 = 7 + (9 + 3) = (7 + 3) + 9
If a, b and c are any three Real numbers, a + (b + c) = (a + b) + c = (a + c) + b |
Now, when we combine both the distributive and associative properties, we can "remove brackets" on a series of expressions and then "collect like terms" like this:
– 5(x + 3y – 2z) + 8 (2x – 2y – 1) = – 5x – 15y + 10z + 16x – 16y – 8
now we reorder the terms:
10z + 16x – 5x – 16y – 15y – 8
and when we collect "like terms", we get
10z + 11x – 31y – 8
Generally, we don't bother to reorder the terms, we just collect them from the original order. Remember that x means 1x so x + 5x = 6x.
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Now for the order in which to perform the operations.
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3. The Order of Operations
When we perform a series of different operations on an expression or in an equation, we must do so in a precisely defined order.
Simplify 5 [ 2x + 3(x^{2} – 7x + 6) ] + 3 [– 2x – 4 (x^{2} – 8x + 1) – 7 ]
1st: multiply the inner parentheses ( ) by their coefficients + 3 and – 4.
5 [2x + 3x^{2} – 21x + 18] + 3 [– 2x – 4x^{2} + 32x – 4 – 7] =
2nd: collect like terms within the brackets [ ].
5 [3x^{2} – 19x + 18] + 3[– 4x^{2} + 30x – 11] =
3rd: multiply the brackets [ ] by 5 and by 3, and collect like terms.
15x^{2} – 95x + 90 – 12x^{2} + 90x – 33 =
Finally, collect "like" terms
3x^{2} – 5x + 57
Notice that we remove the inner brackets before the outer ones. This is a common approach in math. We have to do the same thing when composing functions.
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Examples
a) 5(– x² + 3x) – 7(x^{2} – 3x + 1) = – 5x^{ 2} + 15x – 7x^{ 2 } + 21x – 7 = – 12x^{ 2} + 36x – 7 |
a) distribute the multiplication by 5 and – 7 collect like terms |
b) (x + 2)( x – 3) = x( x – 3) + 2( x – 3) = x^{2} – 3x + 2x – 6 = x^{2} – x – 6 |
b) distribute the multiplication distribute the multiplication collect like terms |
c) (a^{ 2} + 2a) + 6a^{ 2} = (a ^{2} + 6a^{ 2} ) + 2a = 7a^{ 2} + 2a |
c) associate like terms collect them |
d) – 2 [3(x^{ 2} + 2x +1)^{ }+ 4(x^{ 2} – 3)] + 6x – 5 = – 2[3x^{ 2} + 6x + 3 + 4x^{ 2} – 12] + 6x – 5 = – 2[7x^{ 2} + 6x – 9] + 6x – 5 = – 14x^{ 2} – 12x + 18 + 6x – 5 = – 14x^{ 2} – 6x + 13 |
d) multiply 3(x^{ 2} + 2x + 1), multiply 4(x^{2} – 3) collect like terms in brackets [ ] multiply – 2[ 7x^{ 2} + 6x – 9] collect like terms |
e) x^{ }(6x^{ 4} – 5x³ + x²) – (3x^{ 3 })^{ 2} + 1 = 6x^{ 5} – 5x^{ 4} + x^{ 3} – (9x^{ 6 }) + 1 = – 9x^{ 6} + 6x^{ 5} – 5x^{ 4} + x^{ 3} + 1 |
e) distribute multiplication by x, square (3x^{3}) order in descending powers |
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1. Simplify
a) 2(n + 1) – 3(6n – 5) = | b) – 5(3n + 12) + 7(n – 1) = | c) a(x + y) – 3a(6y – 5x) = |
2. Simplify these expressions:
a) 3(2x + 3) – [5(x^{2} – 2x – 3) + (4x – 1)] + 2x(3x – 2)
b) (3x – 1)(x + 4) – 3[(x + 7)(x – 1) + 5(x^{2} – 2x – 1)]
c) + x (4x + 2) – ½ x (6x)
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1. Simplify
a) 2(n + 1) – 3(6n – 5) = 2n + 2 – 18n + 15 = – 16n + 17 |
b) – 5(3n + 12) + 7(n – 1) = – 15n – 60 + 7n – 7 = – 8n – 67 |
c) a(x + y) – 3a(6y – 5x) = ax + ay – 18ay + 15ax = 16ax – 17ay |
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2. Simplify these expressions
a) 3(2x + 3) – [5(x^{2} – 2x – 3) + (4x – 1)] + 2x(3x – 2)
6x + 9 – [5x^{2} – 10x – 15 + 4x – 1] + 6x^{2} – 4x =
6x + 9 – [5x² – 6x – 16] + 6x^{2} – 4x =
x² + 8x + 25
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b) (3x – 1)(x + 4) – 3[(x + 7)(x – 1) + 5(x^{2} – 2x – 1)] =
3x^{2} + 11x – 4 – 3[x^{2} + 6x – 7 + 5x^{2} – 10x – 5] =
3x^{2} + 11x – 4 – 3[6x^{2} – 4x – 12] =
3x^{2} + 11x – 4 – 18x^{2} + 12x + 36 =
– 15x^{2} + 23x + 32
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c)
5x^{2} – 3x + 7 + 4x^{2} + 2x – 3x^{2} = 6x^{2} – x + 7
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(all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).