Download 1 Chapter 1 REVIEW OF FUNDAMENTAL OPERATIONS

Chapter 1
REVIEW OF FUNDAMENTAL OPERATIONS
THE REAL NUMBER SYSTEM
The Real Number System – is a set of elements consist of real numbers. Real numbers include positive numbers, negative
numbers and zero and can be classified as rational or irrational.
Rational Numbers – a number which can be expressed as a quotient or ratio of two integers.
Irrational Numbers - a number which cannot be expressed as a quotient of two integers.
REAL NUMBERS
RATIONAL NUMBERS
INTEGERS
Negative
DECIMALS
Non-Negative/Whole
Numbers
Zero
IRRATIONAL NUMBERS
Fig. 1.
The Real Number System
Positive
DEFINITION OF TERMS
The Absolute Value of a Number – the magnitude of the number disregarding its sign. It corresponds to the distance of the point
associated with the given number from the origin with reference to the number scale.
Example:
| -4 | = 4
|5|=5
|0|=0
Literal Numbers – was brought about in the third century by the Father of Algebra, Diophantus, the Greek algebraist. The use of letters to
represent numbers greatly facilitated the presentation and notation of rules and algebraic processes.
Constant – a symbol that assumes one specific value.
Variable - a symbol that assumes many values.
Algebraic Expression – the result of performing one or a combination of operations of addition, subtraction, multiplication or
division on literal numbers. It is a collection of constants and variables involving at least one of the basic operations in
mathematics.
Term – an expression preceded by plus or minus sign.
Example:
The following are examples of algebraic expression of:
one term:
r2 , 2/x , (a+b)/(a-b), √y , (x-y)4
two terms:
ab –3, [(x-abc)/3] - 1 , 1/x – 1/y
Monomial – an expression consisting of one term.
Binomial – an expression of two terms.
Polynomial / Multinomial – an expression of two or more terms.
Numerical Coefficient – the numerical factor (or constant) in a product.
Literal Coeficient – the literal factor (or variable) in a product.
Similar or Like Terms – term which have the same literal coefficients or factors.
Degree of a Term – (of one variable) is the exponent of that variable; (of two variables) is the sum of the exponents of these two variables.
Degree of an Algebraic Expression – is the highest degree of any term in the expression.
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THE FUNDAMENTAL LAWS
The fundamental operations – addition, subtraction, multiplication and division are governed by five fundamental laws.
The Commutative Law of Addition – states that the order of the addends does not affect the sum.
a+b=b+a
The Commutative
Commutative Law of Multiplication – states that the order of the factors does not affect the product.
ab = ba
The Associative Law of Addition – the sum of three or more numbers does not change in whatever way the addends are
grouped together.
a + b + c = (a + b) + c = a + (b + c) = (a + c) + b
The Associative Law of Multiplication – the product of three or more numbers is the same regardless of the way the factors
are grouped together.
abc = (ab)c = a(bc) = (ac)b
The Distributive Law – Multiplication is distributive with respect to addition. It states that the product of a number and the
sum of other numbers is the same as the sum of the products obtained by multiplying each of the other numbers by the
first number.
a(b + c) = ab + ac
ADDITION & SUBTRACTION OF SIGNED NUMBERS
•
•
•
To add two numbers having like signs, find the sum of their absolute values and prefix the common sign.
To add two numbers having unlike signs, subtract their absolute values and prefix the sign of the number with the greater absolute
value.
To subtract signed numbers, changed the sign of the subtrahend and proceed as in addition.
Examples:
Addition:
Subtraction:
1. (-4) + (-5) = -9
1. (-4) - (-5) = (-4) + 5 = 1
2. 4 + 5 = 9
2. 4 - 5 = 4 + (-5) = -1
3. (-4) + 5 = 1
3. (-4) - 5 = (-4) + (-5) = -9
4. 4 + (-5) = -1
4. 4 - (-5) = 4 + 5 = 9
MULTIPLICATION OF SIGNED NUMBERS
•
•
•
To multiply two numbers having like signs, find the product of their absolute values and prefix the plus (+) sign.
To multiply two numbers having unlike signs, find the product of their absolute values and prefix the negative (-) sign.
To multiply two or more signed numbers, find the product of their absolute values and prefix the plus (+) or minus (-) sign depending on
whether the number of negative factors is even or odd respectively.
Examples:
1. (-4)(-5) = 20
2. ( 4)( 5) = 20
3. (-4)( 5) = -20
4. ( 4)(-5) = -20
DIVISION OF SIGNED NUMBERS
•
•
•
To divide numbers with like signs, divide their absolute value and prefix the positive sign.
To divide numbers with unlike signs, divide their absolute values and prefix the negative sign.
The ratio of a number to itself is unity.
Examples:
1. -20 / -5 = 4
2. 20 / 5 = 4
3.
4.
-20 / 5 = -4
20 / -5 = -4
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COMBINING SIMILAR TERMS
•
•
The process of adding or subtracting like terms. Only similar terms can be combined; the sum of two unlike terms can only
be indicated as a sum.
To combine two or more similar terms, add or subtract the numerical coefficients and affix the common literal factor. This
follows from the distributive law.
Examples:
1. 7x – 5x = (7 – 5)x = 2x
2. 3ab + 4ab = (3 + 4)ab = 7ab
3. 2xy + 5xy - 4xy = (2 + 5 – 4)xy = 3xy
ADDITION & SUBTRACTION OF POLYNOMIALS
•
To add (or subtract) polynomials, arrange similar terms in columns and find the sum (or difference) of the terms in each
column.
Examples:
1. Combine 9x – 3y + 4z, 5y – 3x –7z; 6z + 5x – 4y
2. Subtract x3 + 3xy2 – 2x2y + y3 from 2x3 – y3 + 2xy2
(arranging terms)
+ 2xy2 – y3
arranging similar terms in columns
2x3
x3 – 2x2y + 3xy2 + y3
9x – 3y + 4z
– 3x + 5y – 7z
(changing the sign of the subtrahend)
5x – 4y + 6z
2xy2 – y3
2x3
3
2
-x + 2x y - 3xy2 - y3
sum
11x—
11x—2y + 3z
(proceeding in addition)
x3 + 2x2y - xy2 - 2y3
(difference)
REMOVAL OR INSERTION OF GROUPING SYMBOLS
Addition of algebraic expressions frequently involves the symbols of grouping such as parentheses ( ), bracket [ ] and
braces { }.
• In removing a parenthesis preceded by a positive sign, the sign of each of the terms enclosed is not changed.
• In removing a parenthesis preceded by a negative sign, the sign of each of the terms enclosed is changed.
• When more than one grouping symbol is used, the innermost parenthesis are removed first, working outwards at each subsequent step.
Examples:
1. - ( - 2 + 5x—7y) = 2—5x + 7y
2. 2a + [3a - 2(a - 2b) - 3a]
= 2a + [3a –2a +4b –3a]
= 2a + [-2a + 4b]
= 2a –2a + 4b
=
4b
3. - {a - [a - (2a - 7)]}
= - {a -[a –2a +7]}
= - {a -[-a +7]}
= - {a + a –7}
= - {2a –7}
= -2a + 7
POWERS & EXPONENTS
A compact notation for the product of n factors each of which is a is given in the following definition.
The power a is defined as follows:
an = a · … · a · … · a, n is positive integer
n factors
n is the exponent , a is the base.
n
The Laws of Exponents
•
The Product of Powers
am · an = am+n
Examples:
1.
2.
3.
4.
3
2
3+2
5
a · a =a
= a
2
4
5
2+4+5
11
x · x ·x = x
= x
3
5
8
2 · 2 = 2
3
4
7
(2x )(-3x ) = -6x
3
•
The Quotient of Powers
Examples:
7
am
2
5
1. x / x = x
2
2
2. –y / y = -1
6
10
4
3. x / x = 1/x
2 2
2 3
4. –2 r / (-2) r = -1/r
2
5
2 15
4. 8m nk / 12mn k
10
= 2m/3nk
an
am
an
am
a
n
= a m −n ; if m > n
=1
; if m = n
1
=
a
n −m
; if m < n
Examples:
•
1.
2.
3.
4.
5.
The Power of a Power
(am)n = amn
•
(x4)3 = x12
(3x4)3 = 27x12
(-4m2n8)2 = 16m4n16
-3(m4)3 = -3m12
-52(a4b3c2)5 = -25a20b15c10
Examples:
The Power of a Product
(ab)m
=
1.
2.
3.
4.
5.
6.
ambm
(3m)3 = 33m3 = 27m3
(2x3y2z4)4 = 16x12y8z16
–3(y3)2 = -3y6
(-3y4)2 = 9y8
–32(a3b2c4)2 = -9a6b4c8
–(32a3b2c4)2 = -81a6b4c8
Examples:
1.
[(-3x2) / 2]2 = (-3)2x4 / 22 =
9x4 / 4
2. (3x2y / y3)4 = 34x8y4 / y12 =
81x8 / y8
3. – (15a3x5 / 10ax9)2
= - (3a2 / 2x4)2
= - 9a4 / 4x4
•
The Power of a Quotient
(a / b)m = am / bm
MULTIPLICATION OF POLYNOMIALS
•
•
To multiply two monomials, use commutative, associative and the laws of exponents in multiplication.
Example:
(4x2y3w5) (-8x2y4z3wv)
= 4(-8)(x2 x2)(y3y4) (w5w)(v)
= -32x4y7w6v
= -32vw6x4y7
To multiply two polynomials, use the distributive law and apply the previous rule.
Examples:
1. (3x – 4y)(5x – 2y)
= 3x(5x – 2y) –4y(5x – 2y)
= 15x2 – 6xy – 20xy + 8y2
= 5x2 – 26xy + 8y2
2. (4x2 – 2y)(x2 – 3xy + 5y2)
= 4x2(x2 – 3xy + 5y2) – 2y(x2 – 3xy + 5y2)
= 4x4-12x3y+20x2y2-2x2y+6xy2-10y3
= 4x4-12x3y+20x2y2-2x2y+6xy2-10y3
DIVISION OF POLYNOMIALS
•
The quotient of a monomial divided by another monomial is a monomial with a numerical coefficient equal to the quotient
of the numerical coefficients and a literal coefficient equal to the quotient of the literal coefficients, observing the rule of
signs for division and the exponential laws for multiplication and division.
Examples:
1. x5y6 ÷ x7y3 = x5 – 7 y6 – 3 = x-2y3 = y3/x2
2. 12a4b7 ÷ 36a5b7 = 1/ 3a
3. 72m6n7 ÷ 18m3n5 = 4m3n2
•
To divide one polynomial by another:
1.
2.
Arrange both the dividend and the divisor in decreasing power of one variable.
Divide the first term of the dividend by the first term of the divisor; the result obtained will be referred to as a partial
quotient.
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3.
Multiply the partial quotient obtained in step (2) by all the terms of the divisor and subtract the result from the
dividend. The remainder is the new dividend.
4. Repeat steps (2) and (3) until the new dividend (remainder) is either zero or of a degree lower than the first term of the
divisor.
5. The complete quotient consists of the partial quotients plus the remainder indicated as a ratio to the divisor.
Example 1: Divide by x – 1
Solution: The dividend arranged in descending powers of x becomes x – 3x + 3x + 1
The quotient is written as x – 2x + 1 + (2/x-1)
3
2
2
2
x – 2x + 1
x–1
3
2
x – 3x + 3x + 1
3
2
x – x
2
- 2x + 3x
2
- 2x + 2x
x+1
x–1
2
EXERCISES
A. Subtract the sum of the second and third expressions from the first expression:
12x – y – 10z; 3x – y + 7z
1. 3x – 2y + 8z;
2. 8x – y + 5z;
2x + 3y – 6z;
5x – 3y
B. Subtract the first from the sum of the second and third expressions:
1. 2x – y – 3;
4x + y –5;
x + 11y – 8
2. x – 3y + 9;
3x – 7y + 2;
3x + 5y – 6
C. Simplify the following by removing the grouping symbols and combining similar terms:
1. –[4x – (3y – 16x)] – (3x – 2y)
2. –{4y – [13x – (14y – 2x)] – 18y}
3. –{8x – [14y – (5 – 6x) – 2]}
4. 2x – {-3 – [(2x – y – 3) – 3] –5x – y} – 2y
5. –{16x – [6x – (10y – 12x)] – 3}
6. 10a – [3a + 2(b – 3c) – 6]
7. 9{a – [2b –7(a – c)]} – 6[3(a – b) – 4(a – c)]
8. a(b – c) + b(c – a) + c(a – b)
9. 3x + 2[5x – 6y – (x + y – z)]
10. 2{7a2 – [a2 + b2 – (3a2 + b2 – 5)] + c2 + 6}
D. Find the given value of each of the following expressions for the given values of the variables:
1. 5(2x – 3y + 7) – 4{x – 2y[2x – 3 + 4(1 – x)] – xy};
x = 2, y = 3
x = -1
2. –x5 – 4x2 – 8x – 1;
x = -2
3. (x3 – 4x2 + 8x –1) / (-x2 – 7x +5);
4. (x4 –1) / (x2 – 1);
x = -1
5. 5{2a – b + c – 3[2a - |b – c| - 2(a – c)]};
a = 3, b = 2, c = 3
E. Find the products of the following polynomials:
1. 3x2 + 6x – 5; x + 7
2. 8x2 – 7x –9; x – 5
3. 6x2 – x + 12; 3x – 5
4. 9x2 – 4x – 3; 7x + 8
5. 2x3 – 5x2 + 6x – 2; x2 + 2x – 5
6. 2x4 – 3x2 + 5x – 4; 2x3 – 5x –1
7. 2x2 – x3 – 4x + 5; 3x – 2x2 + 7
F. Square (Multiply the expression by itself):
1. 3a2 – 2b2
2. 2x2 – 3x + 5
3. a – b + c
G. Cube (Multiply the expression three times):
1. a + b
2. x + 2y
3. 2x2 – 5y2
H. Perform the indicated operations:
1. (a + b) (a – b)2
2. ( x – y)3 ( x + y)
3. (3x + 2y)2 (2x – 3y)2
I.
Divide the first polynomial by the second:
1. x3 + 5x2 – 8x – 12; x – 2
2. 12x4 – 25x3 + 18x2 + 7x – 20; 3x – 4
3. 8x4 – 20x2 + 3x + 28; 4x2 – 6x – 7
4. x3 – x4 + x2 – x + 2688; x + 7
5. a5 + 32b5; a + 2b
J. Perform the following operations:
1. [(x3 + y3) ÷ ( x + y)] (x – y)
2. (x6 – y6) ÷ [(x + y)(x – y)]
3. (x6 + y6) ÷ [(x + y)(x – y)]
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