Download Chapter 1 THE INTEGERS

THE INTEGERS
1-1 Whole Numbers, Integers, and the Number Line
Integers
..., 3, 2, 1,0,1, 2,3,...
Whole numbers
0,1, 2,3, 4,...
Counting numbers
1, 2,3, 4,...
1-2 Writing and Solving Number Sentences
- Add or subtract the same value from both sides of the equation
- Multiply or divide by the same value on both side of the equation
** For inequalities multiplying or dividing by a negative number changes the
direction of the inequality symbol.
1-3 Adding Polynomials
Combine like terms
**Remember the variables and the exponents much be the same:
a 2  2a 2b  2a 2  3a 2  2a 2b
Ex:
2t 3  2t 2  4t 3  6t 3  2t 2
1-4 Solving Absolute Value Equations and Inequalities
If x  3 then x  3
For an absolute value problem set the value in the absolute value symbol equal to the positive and
negative value on the other side of the equal sign. Then solve both equations.
Ex: x  3  7
x3 7
x4
Ex 2: 2x  3  x 1
or
x  3  7
x  10
2x  3  x 1
x  4
or
2 x  3  ( x  1)
2x  3  x 1
3 x  2
2
x
3
For absolute value inequalities, if the equation is an absolute value less than a value the solution is
an AND statement.
Ex: x  3  7
x3 7
x4
AND
x  3  7
x  10
10  x  4
which means that x is between -10 and 4.
** Note the change in the inequality symbols:
- Greater than the negative value, less then the positive value.
If the absolute value inequality is an absolute value greater than a value, the solution is an OR
statement.
Ex: x  3  7
x3 7
x4
OR
x  3  7
x  10
** Again, note the change in the inequality symbols:
- Greater than the positive value, less then the negative value.
1-5 Multiplying Polynomials
x m  x n  x mn
When multiplying polynomials multiply the coefficients and add the exponents when the bases are
the same.
** Remember – when no exponent is shown the exponent equals 1
Ex: (2 x 2 y3 )(3xy 2 )  6 x3 y 5
Use the distributive property when multiplying a monomial and a polynomial.
Ex: (2 x2 y3 )(3xy 2  2 x2 y)  (2 x2 y3 )(3xy 2 )  (2 x2 y3 )(2 x2 y)
 6 x3 y 5  4 x 4 y 4
When multiplying 2 binomials distribute both terms in the first binomial to both terms in the second
binomial. Then combine like terms.
*Remember FOIL – First Outside Inside Last
Ex:  2 x  3 y  3x  y    2 x 3x  y   3 y 3x  y 
 6 x 2  2 xy  9 xy  3 y 2
 6 x 2  11xy  3 y 2
When multiplying a polynomial by a polynomial distribute all terms in the first binomial to all terms
in the second binomial.
Ex:  2x  3 y  4 z  3x  y  2z    2x 3x  y  2z   3 y 3x  y  2z    4z 3x  y  2z 
 6 x 2  2 xy  4 xz  9 xy  3 y 2  6 yz  12 xz  4 yz  8 z 2
 6 x2  11xy  3 y 2  8xz  2 yz  8 z 2
1-6 Factoring Polynomials
When factoring, look for:
1. Great Common Factor (monomial or binomial)
2. Difference of 2 squares
3. Trinomials
1-7 Quadratic Equations with Integral Roots
To solve a quadratic equation:
1. Write the equation in standard form.
ax 2  bx  c  0 where a is a positive number.
2. Factor.
3. Set each factor equal to zero.
4. Solve for x.
1-8 Quadratic Inequalities
If ab  0 , then a and b have to have the same sign, because a positive times a positive is a
positive and a negative times a negative is a positive.
If ab  0 , then a and b have to have opposite signs, because a positive times a negative or a
negative times positive is a negative.
When solving a quadratic inequality both cases must be tested.
Ex:
x2  2x  3  0
 x  3 ( x 1)  0
x  3  0 and x 1  0
x  3
x 1
or
x  3  0 and x 1  0
x  3
x 1