may be added (or subtracted, of course)
Chapter 4: Graphing and Straight Lines 65 4.
Nothing else is permitted! This means that
any equation containing things like x 2 , y 2 ,
1/x, xy, square roots, or any other function of x
or y is not linear. DESCRIBIN
4 3) (6 4) 2 4 3 6 4 2 2 2 = + - + - + - = + - + - x x x
x x x x x Notice that the parentheses in the first
line are only there to distinguish the two
polynomials. Although this is basically just a
bookkeeping activity, it can get a little messy
when ther
Example: Given equation: 2 2 1 2 1 2 1 x x x x x
- + - - + Factor both denominators: ( 1)( 1) ( 1)
( 1 2 ) 1 x x x x x x - + + - - - Assemble the LCD:
Note that the LCD contains both denominators
LCD = (x x + - 1)( 1)(x -1) 2 1 ( 1)( 1)( 1) x LCD x x
x -
Write as a binomial squared: (the constant in
the binomial is half the coefficient of x) ( ) 16
25 2 4 3 x + = Square root both sides:
(remember to use plus-or-minus) 4 5 4 3 x + =
Solve for x: 4 - 3 5 x = Chapter 9: Quadratic
Equations 119 Thus x = or x
common factors from each group: x(4x 5) +
3(4x 5) Notice that the two quantities in
parentheses are now identical. That means we
can factor out a common factor of (4x - 5): (4x
5)(x + 3) The Procedure Given a general
quadratic trinomial ax 2 + bx + c 1.
you need to factor a trinomial such as 2x 2 + x 3, you have to think about what combinations
could give the 2x 2 as well as the other two
terms. In this example the 2x 2 must come
from (x)(2x), and the constant term might
come from either (-1)(3) or (1)(-
like 2.7, 3.14, etc. and added them to our
graph. Eventually the points would be so
crowded together that they would form a solid
line: x y Chapter 4: Graphing and Straight Lines
64 The arrows on the ends of the line indicate
that it goes on forever, beca
term (mono means one). The following are
monomials: x 3x 4 2x 3 BINOMIAL A binomial
has two terms: Chapter 6: Polynomials 82 x + 1
5x 2 3x TRINOMIAL A trinomial has three
terms: x 4 + 2x 3 3x 2x 2 4x + 1 DEGREE OF
A TERM The degree of an individual term i
and relationships became more complicated,
their verbal descriptions became harder and
harder to understand. Our modern algebraic
notation greatly simplifies this task. A wellknown formula, due to Einstein, states that E =
mc2 . This remarkable formula gi
noticing that there is a factor common to both
the numerator and the denominator (a factor
of 2 in this example), which we can divide out
of both the numerator and the denominator. 2
3 2 2 3 2 6 4 = = We use exactly the same
procedure to reduce rational e
x x x x - = - - = - - - Note: If you
are dealing with an nth root instead of a
square root, then you need n factors of that
root in order to make it go away. For instance,
if it is a cube root (n = 3), then you need to
multiply by two more factors of that
4ac = 0 There is one real root If b 2 4ac < 0
There are no real roots Chapter 9: Quadratic
Equations 120 DERIVING THE QUADRATIC
FORMULA The quadratic formula can be
derived by using the technique of completing
the square on the general quadratic formula:
neither positive nor negative. WARNING: Do
not attempt to do something like the
distributive law with radicals: a + b a + b
(WRONG) or a + b a + b 2 2 (WRONG). This is
a violation of the order of operations. The
radical operates on the result of everythin
Chapter 5: Systems of Linear Equations THE
SOLUTIONS OF A SYSTEM OF EQUATIONS A
system of equations refers to a number of
equations with an equal number of variables.
We will only look at the case of two linear
equations in two unknowns. The situation get
indeed it can be if the numbers you are
working with have a lot of factors, but in
practice you usually only have to try a few
combinations before you see what will work.
As a demonstration, lets see how we would
attack the example by this method. Given 2
If a quadratic equation has no constant term
(i.e. c = 0) then it can easily be solved by
factoring out the common x from the
remaining two terms: ( ) 0 0 2 + = + = x ax b ax
bx Then, using the zero-product rule, you set
each factor equal to zero and solv
of multiplying polynomials. Recall that when
we factor a number, we are looking for prime
factors that multiply together to give the
number; for example 6 = 2 3 , or 12 = 2 2 3.
When we factor a polynomial, we are looking
for simpler polynomials that can
you have it is still just an approximation of the
exact solution. In real life, though, a close
approximation is often good enough. SOLVING
BY SQUARE ROOTS NO FIRST-DEGREE TERM If
the quadratic has no linear, or first-degree
term (i.e. b = 0), then it can
example: Example: 2y + x = 3 (1) 4y 3x = 1 (2)
Equation 1 looks like it would be easy to solve
for x, so we take it and isolate x: 2y + x = 3 x =
3 2y (3) Now we can use this result and
substitute 3 - 2y in for x in equation 2: ( ) 1 10
10 10 9 1 4 9 6 1
TRINOMIALS (QUADRATIC) A quadratic
trinomial has the form ax 2 + bx + c, where the
coefficients a, b, and c, are real numbers (for
simplicity we will only use integers, but in real
life they could be any real number). We are
interested here in factoring q
We know the two rates, and we know that the
difference between the two distances traveled
will be one mile, but we dont know the actual
distances. Let D be the distance that you travel
in time t, and D + 1 be the distance that the
other car traveled in ti
example. If a is not 1 then things get a little bit
more complicated, so we will begin by looking
at a = 1 examples. Chapter 6: Polynomials 89
Coefficient of x 2 is 1 Since the trinomial comes
from multiplying two first-degree binomials,
lets review what
a fixed point, called the origin, and are
measured according to the distance along a
pair of axes. The x and y axes are just like the
number line, with positive distances to the
right and negative to the left in the case of the
x axis, and positive distan
coefficient of x is 2, and the additive constant
is 1. This is not a coincidence, but is due to
the standard form in which the equation was
written. Standard Form (Slope-Intercept Form)
Chapter 4: Graphing and Straight Lines 71 If a
linear equation in two
mind as you work with rational expressions,
because it can happen that you are trying to
solve an equation and you get one of the
forbidden values as a solution. You would
have to discard that solution as being
unacceptable. You can also get some crazy
re
special products of binomials that 2 2 2 (a + b)
= a + 2ab + b and 2 2 2 (a - b) = a - 2ab + b The
trinomials on the right are called perfect
squares because they are the squares of a
single binomial, rather than the product of two
different binomials. A
identified by the unique combination of
positive and negative signs for the coordinates
of a point in that quadrant. x y I (+,+) II (-,+) III
(-,-) IV (+,-) Chapter 4: Graphing and Straight
Lines 62 GRAPHING FUNCTIONS Consider an
equation such as y = 2x 1
- x + 6 This method can get hard to keep track
of when there are many terms. There is,
however, a more systematic method based on
the stacked method of multiplying numbers:
Stack the factors, keeping like degree terms
lined up vertically: 2 2 3 2 x x x -
Centre of Excellence for Education in
Mathematics (ICE-EM), the education division
of the Australian Mathematical Sciences
Institute (AMSI), 2010 (except where
otherwise indicated). This work is licensed
under the Creative Commons
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