...John Naisbitt
In just FIVE minutes you should learn to quickly multiply any two numbers from 11 to 19 in your head.
Try this:
- Take 15 x 13 for an example.
- Always place the larger number of the two on top in your mind.
- Then draw the shape of Africa mentally so it covers the 15 and the 3 from the 13 below. Those covered numbers are all you need.
- First add 15 + 3 = 18
- Add a zero behind it (multiply by 10) to get 180.
- Multiply the covered lower 3 x the single digit above it the "5" (3x5= 15)
- Add 180 + 15 = 195.
1.1 Sets of Numbers and Interval Notation
1.2 Operations on Real Numbers
1.3 Simplifying Expressions
1.4 Linear Equations in One Variable
1.5 Applications of Linear Equations in One Variable
1.6 Literal Equations and Applications to Geometry
1.7 Linear Inequalities in One Variable
1.8 Properties of Integer Exponents and Scientific Notation
2.1 The Rectangular Coordinate System and Midpoint Formula
2.2 Linear Equations in Two Variables
2.3 Slope of a Line
2.4 Equations of a Line
2.5 Applications of Linear Equations and Graphing
3.1 Solving Systems of Linear Equations by Graphing
3.2 Solving Systems of Equations by Using the Substitution Method
3.3 Solving Systems of Equations by Using the Addition Method
3.4 Applications of Systems of Linear Equations in Two Variables
3.5 Systems of Linear Equations in Three Variables and Applications
3.6 Solving Systems of Linear Equations by Using Matrices
3.7 Determinants and Cramer�s Rule
4.1 Introduction to Relations
4.2 Introduction to Functions
4.3 Graphs of Functions
4.4 Variation
5.1 Addition and Subtraction of Polynomials and Polynomial Functions
5.2 Multiplication of Polynomials
5.3 Division of Polynomials Problem Recognition Exercises�Operations on Polynomials
5.4 Greatest Common Factor and Factoring by Grouping
5.5 Factoring Trinomials
5.6 Factoring Binomials
5.7 Additional Factoring Summary
5.8 Solving Equations by Using the Zero Product Rule
6.1 Rational Expressions and Rational Functions
6.2 Multiplication and Division of Rational Expressions
6.3 Addition and Subtraction of Rational Expressions
6.4 Complex Fractions Problem Recognition Exercises�Operations on Rational Expressions
6.5 Rational Equations
6.6 Applications of Rational Equations and Proportions
7.2 Rational Exponents
7.3 Simplifying Radical Expressions
7.4 Addition and Subtraction of Radicals
7.5 Multiplication of Radicals
7.6 Rationalization
7.7 Radical Equations
7.8 Complex Numbers
8.2 Quadratic Formula
8.3 Equations in Quadratic Form
8.4 Graphs of Quadratic Functions
8.5 Vertex of a Parabola and Applications
9.2 Polynomial and Rational Inequalities
9.3 Absolute Value Equations
9.4 Absolute Value Inequalities Problem Recognition Exercises�Equations and Inequalities
9.5 Linear Inequalities in Two Variables
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f(x) = q(x)(x - c) + f(c).
Moreover, if f(x) = q1(x)(x - c) + k, where q1(x) F[x] and k F, then q1(x) = q(x) and k = f(c).
4.1.10. Definition. Let f(x) = amxm + � � � + a0 F[x]. An element c F is called a root of the polynomial f(x) if f(c) = 0, that is, if c is a solution of the polynomial equation f(x) = 0 .
4.1.11. Corollary. Let f(x) F[x] be a nonzero polynomial, and let c F. Then c is a root of f(x) if and only if x-c is a factor of f(x). That is,
f(c) = 0 if and only if (x-c) | f(x).
4.1.12 Corollary. A polynomial of degree n with coefficients in the field F has at most n distinct roots in F.
4.2.1. Theorem. [Division Algorithm] For any polynomials f(x) and g(x) in F[x], with g(x) 0, there exist unique polynomials q(x),r(x) F[x] such that
f(x) = q(x)g(x) + r(x),
where either deg(r(x)) < deg(g(x)) or r(x) = 0.
4.2.2. Theorem Let I be a subset of F[x] that satisfies the following conditions:
(i) I contains a nonzero polynomial;
(ii) if f(x),g(x) I, then f(x)+g(x) I;
(iii) if f(x) I and q(x) F[x], then q(x)f(x) I.
If d(x) is any nonzero polynomial in I of minimal degree, then
I = { f(x) F[x] | f(x)=q(x)d(x) for some q(x) F[x] }.
4.2.3. Definition. A monic polynomial d(x) F[x] is called the greatest common divisor of f(x),g(x) F[x] if
(i) d(x) | f(x) and d(x) | g(x) , and
(ii) if h(x) | f(x) and h(x) | g(x) for some h(x) F[x], then h(x) | d(x).
The greatest common divisor of f(x) and g(x) is denoted by gcd(f(x),g(x)).
If gcd(f(x),g(x)) = 1, then the polynomials f(x) and g(x) are said to be relatively prime.
4.2.4. Theorem. For any nonzero polynomials f(x),g(x) F[x] the greatest common divisor gcd(f(x),g(x)) exists and can be expressed as a linear combination of f(x) and g(x), in the form
gcd(f(x),g(x)) = a(x)f(x) + b(x)g(x)
for some a(x),b(x) F[x].
Example. 4.2.3. (Euclidean algorithm for polynomials) Let f(x),g(x) F[x] be nonzero polynomials. We can use the division algorithm to write
f(x) = q(x)g(x) + r(x),
with deg(r(x))
� If r(x) = 0, then g(x) is a divisor of f(x), and so gcd(f(x),g(x)) = cg(x), for some c F.
� If r(x) 0, then it is easy to check that gcd(f(x),g(x)) = gcd(g(x),r(x)).
This step reduces the degrees of the polynomials involved, and so repeating the procedure leads to the greatest common divisor of the two polynomials in a finite number of steps.
The Euclidean algorithm for polynomials is similar to the Euclidean algorithm for finding the greatest common divisor of nonzero integers. The polynomials a(x) and b(x) for which
gcd(f(x),g(x)) = a(x)f(x) + b(x)g(x)
can be found just as for integers (see the Euclidean algorithm for integers).
4.2.5. Proposition. Let p(x),f(x),g(x) F[x]. If gcd(p(x),f(x)) = 1 and p(x) | f(x)g(x), then p(x) | g(x).
4.2.6. Definition. A nonconstant polynomial (that is, a polynomial with positive degree) is said to be irreducible over the field F if it cannot be factored in F[x] into a product of polynomials of lower degree. It is said to be reducible over F if such a factorization exists.
4.2.7. Proposition. A polynomial of degree 2 or 3 is irreducible over the field F if and only if it has no roots in F.
4.2.8 Lemma. The nonconstant polynomial p(x) F[x] is irreducible over F if and only if for all f(x),g(x) F[x],
p(x) | (f(x)g(x)) implies p(x) | f(x) or p(x) | g(x).
4.2.9. Theorem. [Unique Factorization] Any nonconstant polynomial with coefficients in the field F can be expressed as an element of F times a product of monic polynomials, each of which is irreducible over the field F . This expression is unique except for the order in which the factors occur.
4.2.10. Definition. Let f(x) F[x]. An element c F is said to be a root of multiplicity n 1 of f(x) if
(x - c)n | f(x) but (x - c)n+1 f(x).
4.2.11. Proposition. A nonconstant polynomial f(x) over the field R of real numbers has no repeated factors if and only if gcd(f(x),f'(x))=1, where f'(x) is the derivative of f(x).
Example. 4.2.4. (Partial fractions) Let f(x)/g(x) be a rational function. The first step in achieving a partial fraction decomposition of f(x)/g(x) is to use Theorem 4.2.9to write g(x) as a product of irreducible polynomials.
If g(x)=p(x)q(x), where p(x) and q(x) are relatively prime, then by Theorem 4.2.4 there exist polynomials a(x) and b(x) with
a(x)p(x)+b(x)q(x)=1.
Dividing by p(x)q(x) allows us to write
1 / p(x)q(x) = a(x)/q(x) + b(x)/p(x),
and so
f(x) / g(x) = (f(x)a(x)) / q(x) + (f(x)b(x)) / p(x).
This process can be extended by induction until f(x)/g(x) is written as a sum of rational functions, where in each case the denominator is a power of an irreducible polynomial.
The next step in the partial fraction decomposition is to expand the terms of the form h(x)/p(x)n. Using the division algorithm, we can write
h(x) / p(x)n = a(x) + r(x)/p(x)n,
where deg(r(x)) < deg(p(x)n). Then we can divide r(x) by p(x)n-1 to obtain
r(x) = b(x)p(x)n-1 + c(x),
where deg(c(x)) < deg(p(x)n-1). This gives us
r(x) / p(x)n = b(x)/p(x) + c(x)/p(x)n,
in which deg(b(x)) < deg(p(x)). This can be continued by induction, to obtain
h(x) / p(x)n= a(x) + b(x)/p(x) + � � � + t(x)/p(x)n,
in which the numerators b(x),...,t(x) all have lower degree than that of p(x).
Construction of extension fields
4.4.1. Definition. Let E and F be fields. If F is a subset of E and has the operations of addition and multiplication induced by E, then F is called a subfield of E, and E is called an extension field of F.
4.4.2. Definition. Let F be a field, and let p(x) be a fixed polynomial over F. If a(x),b(x) F[x], then we say that a(x) and b(x) are congruent modulo p(x), written
a(x) b(x) (mod p(x)),
if p(x) | (a(x)-b(x)). The set
{ b(x) F[x] | a(x) b(x) (mod p(x)) }
is called the congruence class of a(x), and will be denoted by [a(x)].
The set of all congruence classes modulo p(x) will be denoted by
F[x]/
4.4.3. Proposition. Let F be a field, and let p(x) be a nonzero polynomial in F[x]. For any a(x) F[x], the congruence class [a(x)] modulo p(x) contains a unique representative r(x) with deg(r(x))
4.4.4. Proposition. Let F be a field, and let p(x) be a nonzero polynomial in F[x]. For any polynomials a(x),b(x),c(x), and d(x) in F[x], the following conditions hold:
(a) If a(x) c(x) (mod p(x)) and b(x) d(x) (mod p(x)), then
a(x)+b(x) c(x)+d(x) (mod p(x))
and
a(x)b(x) c(x)d(x) (mod p(x)).
(b) If gcd(a(x),p(x))=1, then
a(x)b(x) a(x)c(x) (mod p(x))
implies
b(x) c(x) (mod p(x)).
4.4.5. Proposition. Let F be a field, and let p(x) be a nonzero polynomial in F[x]. For any a(x) F[x], the congruence class [a(x)] has a multiplicative inverse in F[x]/