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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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We write that x2 + 2x is the expansion ofx(x + 2) or (x + 2)x, with x written at the end of the brackets.
The rule for expanding a bracket is that both the terms inside the brackets are multiplied by the term outside the brackets, which is the distributive law.
Example 3. Expand the following brackets: (a) 3x(x + 5), (b) 4(3x — 5),
(c) -3(1 - 2x), (d) x(2x + 3) - 4(x - 6).
Solution. For (a), write
Area = x2 + 2x
Using algebraic shorthand, x x x = x2 and 2 x x = 2x.
This example demonstrates that
x(x + 2) = x2 + 2x
3x(x + 5) = 3x x x + 3x x 5 Using the distributive law = 3.x2 + 15x Using algebraic shorthand
For (b),
4(3x — 5)=4x3x—4x5 The subtraction of terms in brackets is preserved = 12x - 20
EXPANDING BRACKETS 187
(c) If the term in front of the brackets is negative, the signs of all the terms inside the bracket will be changed when they are multiplied by the term in front. Write
—3(1 — 2x) = — 3 + 6x Change in signs: +1 —3, and —2x 4-6x
For (d), write
x (2x + 3) — 4(x — 6) = 2x2 + 3.x — 4.x + 24 Change in signs of the second
brackets
= 2x2 — x + 24 +3.x — 4x = —x
The expansion of double brackets will now be explained using a practical example, which will be followed up with a rule.
Example 4. Suppose Andrew has a square sandpit that measures a: by a: meters, and he decides to extend it 2 m in one direction and 3 m in the other direction. What is the area of the new sandpit?
Solution. The left-hand side of figure c shows the extensions to the sandpit, and the right-hand side shows how the new sandpit is divided up into two rectangles.
X 2 x + 2
X x(x + 2)
3 3(x + 2)
(c)
Using the left-hand side of the figure, write
Area of new sandpit = length x width = (x + 3)(x -+- 2)
Using the right-hand side of the figure, write
Area of new sandpit = area of top rectangle + area of bottom rectangle = x(x + 2) + 3(x + 2)
Since both expressions represent the area of the new sandpit, we can write
(x + 3)(x + 2) = x(x + 2) + 3(x + 2)
= x2 + 2x + 3x + 6 Using the distributive law for each set
of brackets
= x2 -\- Sx -\- 6 Collecting like terms, 4-2x 4- 3x = +5x
188 EXPERIMENTAL PROBABILITY
The rule for expanding double brackets can now be stated:
The second bracket is multiplied by each term in the first bracket, and then the two brackets are expanded.
Worked examples follow that explain how to solve a variety of problems.
Expand
Expand
Expand
Expand
(x — 3)(x + 5) = x(x + 5) — 3(x + 5)
= x2 + 5.x — 3.x — 15
= x2 + 2x — 15 (x — 5)(x + 5) = x(x + 5) — 5(x + 5)
= x2 + 5x — 5x — 25 = x2 — 25 (2x - 3)2 = (2x- 3)(2x - 3)
= 2x(2x - 3) - 3(2x - 3) = 4x2 — 6x — 6x + 9
= 4jc2 - 12x + 9 (2x + y)(3a — b) = 2x(3a — b) + y (3a — b) = 6ax — 2 bx + 3 ay — by
Note the changes in signs
+5x — 3x = +2x
+5x — 5x = 0
Squaring means multiplying by itself
Note the changes in signs
—6x — 6x = — I2x
There are no like terms to collect
References: Algebra, Brackets, Coefficient, Distributive Law, Factorize, Pascal’s Triangle, Squaring.
EXPANSION
Reference: Expanding Brackets.
EXPERIMENT
Reference: Event.
EXPERIMENTAL PROBABILITY
Reference: Probability of an Event.
EXPONENT 189
EXPONENT
When a number or an expression is multiplied by itself many times it is simpler to express the result in exponent form. For example, 81=3x3x3x3 and is written as 34. We say that 34 is the exponent form of3x3x3x3 and its value is 81. When 34 replaces 81, we say that 3 is the base and 4 is the exponent. Another name for exponent is index. The plural of index is indices.
The power is the number of times the base is multiplied by itself. In the example 34 we say that 81 is the fourth power of 3, or 81 is 3 to the power 4. Special names are used for powers of 2 and 3. For example, 42 is four squared or four to the power 2, and 53 is five cubed or five to the power 3. Another way of expressing exponent form is as follows: 5 is raised to the power 4, which means 54.
When numbers or algebraic terms in index form are multiplied or divided certain rales can be applied. The laws of indices are stated here with examples to illustrate their uses. The first law is the exponent form of an expression.
Rule 1:
axaxaxaxaxa... ton terms = an
Note that 6n is not the same as 6n, which means 6 x ft.
Examples of this rale are
23 = 2 x 2 x 2
= 8
y^xy^=yxyxyxyxyxyxy
= y7
a1 = a
Rule 2:
When two terms in exponent form are multiplied the exponents are added together, provided they have the same base.
Examples of this rale are
35 x 32 = 37 Both terms are in base 3
y5 x y4 = y9 Both terms are in base y
(x + 3)2 x (x + 3)4 = (x + 3)7 Both terms are in base (x + 3)
The rule does not apply when the bases are different. For example, we cannot
simplify 34 x 26 using this law of indices, because the bases 3 and 2 are not the same. In the same way y2 x w3 cannot be simplified, because the bases y and w are different.
190 EXPONENT
Rule 3:
CLm
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