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1.08 Polynomials from Intermediate Algebra: One Step at a Time © 1998 Pages 96-102 Dr. Robert J. Rapalje Seminole Community College Under
Construction! Recall
from previous lessons that when algebraic expressions are added
(or subtracted) they are called terms,
while expressions that are multiplied are called factors.
An algebraic expression that contains only one
term is called a monomial. If the expression has two
terms is called a binomial,
and if there are three terms
it is a trinomial.
A polynomial is an
algebraic expression consisting of one or more terms.
A polynomial may consist of numbers and variables, where the
numerical part of a given term is called the coefficient. If there is only one variable in the polynomial, such as x,
then it is called a polynomial in
x. The degree
of a polynomial in one variable is the highest exponent of the
variable. If there is more
than one variable in the polynomial, then the degree
is the highest "sum of the exponents" of the
variables of a given term. Frequently
polynomials can be simplified
by combining like terms; sometimes they can be factored.
Polynomials can be added, subtracted, multiplied (expanded),
or divided. Since addition
and subtraction of polynomials is little more than combining like terms,
and division of polynomials is saved for Chapter 2, this section will
involve only the multiplication
(expansion) of polynomial expressions.
The next section is the factoring
of polynomial expressions, followed immediately by solving
quadratic equations by factoring. Notice
that polynomial expressions
are not equations, and therefore cannot be
"solved." This
chapter involves only polynomial
expressions. This
explanation will begin with a review of products:
monomial times monomial,
monomial times binomial, binomial times binomial, binomial times
trinomial, and trinomial times trinomial.
The basic property that underlies these products is the distributive
property for multiplication (products) over
addition (two or more terms).
Also, the law of exponents about products ("when you
multiply, you add exponents") is used. EXAMPLES: 1.
Monomial times monomial:
4x2 6x3
= 24x5
(by law of exponents) 2.
Monomial times binomial:
4x2 (6x3 + 5x2)
= 24x5 + 20x4
(by distributive property) 3.
Binomial times binomial:
(F OI L):
(3x + 5) (4x + 7) = 12x2 + 41x
+ 35
(This is actually the distributive property applied
twice!) 4.
Binomial times trinomial:
(3x + 5) (4x2 + 7x + 2) =
3x times trinomial: 12x3
+ 21x2 + 6x
5 times trinomial:
20x2
+ 35x + 10
Combine
like terms:
= 12x3 + 41x2 + 41x+
10 5.
Trinomial times trinomial:
(3x2 + 5x + 9) (4x2 +
7x + 2) =
3x2
times trinomial: 12x4 +
21x3 + 6x2
5x times trinomial:
20x3 + 35x2 +
10x
9
times trinomial:
36x2
+ 63x + 18
Combine
like terms:
= 12x4 + 41x3 + 77x2
+ 73x + 18
In
the exercises that follow, these products will be followed by exercises
that reach to a higher level of abstraction.
Hopefully, the "one step" format will help you
understand easily. EXERCISES: 1. (x + 4)(x + 3)
2. (y + 4)(y
+ 3)
3.
[$ + 4][$ + 3] 4. [π + 4][π + 3]
5. [(Junk) +
4][(Junk) + 3] 6.
[(Junk) + 6][(Junk) + 7]
7. [(Junk) + 4][(Junk) - 3]
8.
[(Junk) - 4][(Junk) + 3]
9. [(Junk) -
6][(Junk) - 7]
= x2 + xy +3x
F OI
L
xy
+ y2
+ 3y
= (x
+ y)2 + 7(x + y) +
12
4x
+4y +12
= x2 + 2xy + y2
+ 7x + 7y + 12.
= x2 + 2xy +7x
+y2 +7y + 12. Of
these two methods, the first is the preferred method.
The second may appear easier, but it is not as efficient. Try several problems using the first method; practice other
problems using the second method. Be
familiar with both ways. In 10 - 25, expand the polynomials completely: 10. [(x
+ y) - 4][(x + y) - 3]
11. [(x
+ y) - 4][(x + y) + 7] 12. [(2x + y)
+ 4][(2x + y) - 7]
13. [(x
3y) - 8][(x 3y) - 6] 14.
[(2x-3y) + 8][(2x-3y) - 4]
15.
[(3x-2y) - 8][(3x-2y) + 4]
In
16 - 21, remember that [(x+y)
+ 4]2 = [(x+y) + 4][(x+y) + 4].
16.
[(x + y) + 4]2
17. [(x
+ y) - 4]2
18.
[(3x 5y) + 4]2
19.
[(2x 5y) - 3]2
20.
[(5x 2y) - 8]2
21. [(5x
+ 4y) - 6]2
22.
[(3x-5y) - 4][(3x-5y) + 4]
23. [(2x-5y)
- 3][(2x-5y) + 3]
24.
[(2x+7y) - 5][(2x+7y) + 5]
25. [(3x+8y)
- 7][(3x+8y) + 7]
26.
(x + y)3 = (x + y)
(x + y) (x + y)
= (x + y)
(x2 + 2xy + y2)
=
=
Now
consider the problems (x +
y)3, (x - y)3, (x + y)4,
(x - y)4,
etc. These are binomials
raised to a power. The general case, (x
+ y)n, is called the Binomial
Theorem or Binomial Expansion.
(Pascals Triangle,
the concept of the next two pages may be helpful.
If you find them more confusing than helpful, and you cannot find
someone to convince you of the simplicity of the concept, please
continue to the next section.)
The
following pattern can easily be developed:
(x
+ y)0 =
1
(x
+ y)1 =
1 x +
1
y
(x + y)2 =
1 x2 +
2
xy +
1
y2
(x
+ y)3 = 1
x3
+ 3 x2y
+ 3
xy2
+ 1 y3
1. (x
+ y)4 =
1 x4
+ ____
x3y
+ ____
x2y2
+ ____
xy3
+ 1
y4
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