2
4a
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
Polynomial Identification and Properties of
Exponents
Polynomials
We begin with the definition of a term.
Term. A term is either a single number (called a constant term) or the product
of a number and one or more variables.
For example, each of the following is a term.
−5
− 3x2
12y 2 z 3
13a2 bc3
Note how the first term is a single number, while the remaining terms are
products of a number and one or more variables. For example, −3x2 is the
product of −3, x, and x.
Coefficient. When a term is a product of a number and one or more variables,
the number is called the coefficient of the term. In the case of a term that is
a single number, the number itself is called the coefficient.
Thus, for example, the coefficients of the terms
−5
− 3x2
12y 2 z 3
13a2 bc3
are −5, −3, 12, and 13, respectively.
Degree. The degree of a term is the sum of the exponents on each variable of
the term. A constant term (single number with no variables) has degree zero.
Thus, for example, the degrees of the terms
−5
− 3x2
12y 2 z 3
13a2 bc3
are 0, 2, 5, and 6, respectively. In the last example, note that 13a2 bc3 is
equivalent to 13a2 b1 c3 , so adding exponents, we get:
Degree of 13a2 bc3 = Degree of 13a2 b1 c3
=2+1+3
=6
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
3
Monomial. The words monomial and term are equivalent.
Thus,
− 3x2
−5
12y 2 z 3
13a2 bc3
are monomials.
Binomial. A binomial is a mathematical expression containing exactly two
terms.
The terms in a mathematical expression are separated by plus or minus signs.
For example, each of the mathematical expressions
2x + 3y
− 3a2 − 3b2
xy + 7
− 3x2 y + 5xy 2
is a binomial. Each expression has exactly two terms.
Trinomial. A trinomial is a mathematical expression containing exactly three
terms.
Again, the terms in a mathematical expression are separated by plus or minus
signs. For example, each of the mathematical expressions
2x2 + 3x + 7
a2 + 2ab + b2
x4 − 2x2 y 2 + 3y 4
is a trinomial. Each expression has exactly three terms.
A bicycle has two wheels, a binomial has two terms. A tricycle has three
wheels, a trinomial has three terms. But once we get past three terms, the
assignment of special names ceases and we use the generic word polynomial,
which means “many terms.”
Polynomial. A polynomial is a many-termed mathematical expression. The
coefficients of a polynomial are the coefficients of its terms.
Each of the previous expressions,
12y 2 z 3
− 3a2 − 3b2
x4 − 2x2 y 2 + 3y 4
though assigned the particular names monomial, binomial, and trinomial, respectively, are also “many-termed” expressions and can also be called polynomials. However, because the word polynomial means “many terms,” we can
use the word polynomial to describe mathematical expressions with more than
three terms, such as:
x4 − 4x3 y + 6x2 y 2 − 4xy 3 + y 4
The coefficients of x4 − 4x3 y + 6x2 y 2 − 4xy 3 + y 4 are 1, −4, 6, −4, and 1.
4
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
Ascending and Descending Powers
When asked to simplify a polynomial expression, we should combine any like
terms we find, and when possible, arrange the answer in ascending or descending powers.
You Try It!
EXAMPLE 1. Simplify the following polynomial expression, arranging your
answer in descending powers of x. Once you’ve completed that task, make a
second arrangement, ordering your terms in ascending powers of x.
2x3 + 7x − 3x2 + 11x + 8x2 + 11 + 15x
Solution: In order to arrange our answer in descending powers of x, we want
to place the term with the highest power of x first and the term with the lowest
power of x last. We use the commutative and associative properties to change
the order and regroup, then we combine like terms.
2x3 + 7x − 3x2 + 11x + 8x2 + 11 + 15x
= 2x3 + (−3x2 + 8x2 ) + (7x + 11x + 15x) + 11
= 2x3 + 5x2 + 33x + 11
Note how the powers of x start at 3, then go down in order.
To arrange our final answer in ascending powers of x, we put the lowest
power of x first, then the highest power of x last, regrouping and combining
like terms.
2x3 + 7x − 3x2 + 11x + 8x2 + 11 + 15x
= 11 + (7x + 11x + 15x) + (−3x2 + 8x2 ) + 2x3
= 11 + 33x + 5x2 + 2x3
Note how we start with the constant term, then the powers of x increase in
order.
!
When we have a polynomial in a single variable, such as the polynomial
in Example 1, arranging the terms in ascending or descending order is fairly
straightforward. However, a polynomial in two or more terms is a bit more
difficult, and sometimes impossible, to arrange in a decent order.
You Try It!
EXAMPLE 2. Simplify the following polynomial expression, then arrange
your answer in descending powers of x.
x3 + 2xy 2 − 6x2 y + y 3 − 3xy 2 + 4x2 y
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
5
Solution: We’ll again use the commutative and associative properties to
change the order and regroup, putting the terms with the highest powers of x
first, then follow with terms containing lowers powers of x in order.
x3 + 2xy 2 − 6x2 y + y 3 − 3xy 2 + 4x2 y
= x3 + (−6x2 y + 4x2 y) + (2xy 2 − 3xy 2 ) + y 3
= x3 − 2x2 y − xy 2 + y 3
Note that this is a very natural order, the powers of x decrease while simultaneously the powers of y increase.
!
Not all examples will have nice ordering presented in Example 2, with the
powers of one variable descending while the powers of the other variable simultaneously ascends. Sometimes we have to make some very subjective choices
on the ordering of terms.
You Try It!
EXAMPLE 3. Simplify the following polynomial expression, then arrange
your answer in some sort of reasonable order.
a3 b3 + 2a2 b − 3a2 b3 + 4a3 b3 + 5a4 + 3a2 b + b5
Solution: Let’s try to arrange the terms so that the powers of a descend.
Again, we use the commutative and associative properties to change the order
and regroup.
a3 b3 + 2a2 b − 3a2 b3 + 4a3 b3 + 5a4 + 3a2 b + b5
= 5a4 + (a3 b3 + 4a3 b3 ) + (2a2 b + 3a2 b) − 3a2 b3 + b5
= 5a4 + 5a3 b3 + 5a2 b − 3a2 b3 + b5
Note that in our final arrangement, the powers of a descend, but the powers of
b bounce up and down, but at least we have the powers of a descending. That
should help us spot if we’ve missed a term while simplifying the given problem.
!
The Degree of a Polynomial
To find the degree of a polynomial, locate the term of the polynomial having
the highest degree.
The degree of a polynomial. The degree of a polynomial is the degree of
the term having the highest degree.
6
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
Finding the degree of a polynomial of a single variable is pretty easy.
You Try It!
EXAMPLE 4. What is the degree of the polynomial x3 − 4x2 + 5 − 6x + 2x7?
Solution: First, let’s arrange the polynomial in descending powers of x.
2x7 + x3 − 4x2 − 6x + 5
Arranging the polynomial in descending powers of x makes it easier to see
that the term of the polynomial with the highest degree is 2x7 . Therefore, the
degree of the polynomial is 7.
!
Finding the degree of a polynomial of more than one variable is a little bit
harder.
You Try It!
EXAMPLE 5. What is the degree of the polynomial x4 − 2x3 y 7 + y 5 ?
Solution: Note that the polynomial is already arranged in descending powers
of x, an arrangement that is probably as good as we are going to get. In the
following table, we list the degree of each term. Remember, the degree of any
term is found by summing the exponents on its variables.
Term
4
x
−2x3 y 7
y5
Degree
4
10
5
Hence, the term with the highest degree is −2x3 y 7 , making 10 the degree of
the polynomial.
!
Laws of Exponents
In Chapter 1, section 1, we first introduced the definition of an exponent. For
convenience, we repeat that definition.
In the exponential expression an , the number a is called the base, while the
number n is called the exponent.
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
7
Exponents. Let a be any real number and let n be any whole number. If
n "= 0, then:
an = a
! · a · a"#· · · · · a$
n times
n
That is, to calculate a , write a as a factor n times. In the case where a "= 0,
but n = 0, then we define:
a0 = 1
For example, raising a number to the fifth power requires that we repeat the
number as a factor five times (see Figure ??).
(−2)5 = (−2)(−2)(−2)(−2)(−2)
= −32
Raising a number to the fourth power requires that we repeat that number as
a factor four times.
%
&4 %
&%
&%
&%
&
1
1
1
1
1
−
−
−
−
= −
2
2
2
2
2
1
=
16
As a final example, note that 100 = 1, but 00 is undefined.
Multiplying Like Bases
In the expression an , the number a is called the base and the number n is
called the exponent. Frequently, we’ll be required to multiply two exponential
expressions with like bases, such as x3 · x4 . Recall that the exponent tells us
how many times to write each base as a factor, so we can write:
x3 · x4 = (x · x · x) · (x · x · x · x)
=x·x·x·x·x·x·x
= x7
Note that we are simply counting the number of times that x occurs as a factor.
First, we have three x’s, then four x’s, for a total of seven x’s. However a little
thought tells us that it is much quicker to simply add the exponents to reveal
the total number of times x occurs as a factor.
x3 · x4 = x3+4
= x7
The preceding discussion is an example of the following general law of exponents.
8
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
Multiplying Like Bases. To multiply two exponential expressions with like
bases, repeat the base and add the exponents. In symbols:
am · an = am+n
You Try It!
EXAMPLE 6. Simplify each of the following expressions:
(a) y 4 · y 8
(b) 23 · 25
(c) (x + y)2 (x + y)7
Solution: In each example we have like bases. Thus, the approach will be the
same for each example: repeat the base and add the exponents.
(a) y 4 · y 8 = y 4+8
= y 12
(b) 23 · 25 = 23+5
= 28
(c) (x + y)2 (x + y)7 = (x + y)2+7
= (x + y)9
With a little practice, each of the examples can be simplified mentally. Repeat
the base and add the exponents in your head: y 4 · y 8 = y 12 , 23 · 25 = 28 , and
(x + y)2 (x + y)7 = (x + y)9 .
!
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
9
You Try It!
EXAMPLE 7. Simplify:
(a6 b4 )(a3 b2 )
Solution: We’ll use the commutative and associative properties to change the
order of operation, then repeat the common bases and add the exponents.
(a6 b4 )(a3 b2 ) = a6 b4 a3 b2
The associative property allows us
to regroup in the order we prefer.
= a6 a3 b 4 b 2
The commutative property allows us
to change the order of multiplication.
= a9 b 6
Repeat the common bases and add the
exponents.
With practice, we realize that if all of the operators are multiplication, then we
can multiply in the order we prefer, repeating the common bases and adding
the exponents mentally: (a6 b4 )(a3 b2 ) = a9 b6 .
!
You Try It!
EXAMPLE 8. Simplify:
xn+3 · x3−2n
Solution: Again, we repeat the base and add the exponents.
xn+3 · x3−2n = x(n+3)+(3−2n)
6−n
=x
Repeat the base, add the exponents.
Simplify. Combine like terms.
!
Dividing Like Bases
Like multiplication, we will also be frequently asked to divide exponential expressions with like bases, such as x7 /x4 . Again, the key is to remember that
the exponent tells us how many times to write the base as a factor, so we can
write:
x7
x·x·x·x·x·x·x
=
4
x
x · x · x · x·
x
·x
·x
!
!
!·x
!·x·x·x
=
x
!·x
!·x
! · x·
!
= x3
Note how we cancel four x’s in the numerator for four x’s in the denominator.
However, in a sense we are “subtracting four x’s” from the numerator, so a
10
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
faster way to proceed is to repeat the base and subtract the exponents, as
follows:
x7
= x7−4
x4
= x3
The preceding discussion is an example of the second general law of exponents.
Dividing Like Bases. To divide two exponential expressions with like bases,
repeat the base and subtract the exponents. In symbols:
am
= am−n
an
Note that the subtraction obeys the rule “numerator minus denominator.”
You Try It!
EXAMPLE 9. Simplify each of the following expressions:
(a)
x12
x3
(b)
57
54
(c)
(2x + 1)8
(2x + 1)3
Solution: In each example we have like bases. Thus, the approach will be the
same for each example: repeat the base and subtract the exponents.
(a)
x12
= x12−3
x3
= x9
(b)
57
= 57−4
54
= 53
(c)
(2x + 1)8
= (2x + 1)8−3
(2x + 1)3
= (2x + 1)5
With a little practice, each of the examples can be simplified mentally. Repeat
the base and subtract the exponents in your head: x12 /x3 = x9 , 57 /54 = 53 ,
and (2x + 1)8 /(2x + 1)3 = (2x + 1)5 .
!
You Try It!
EXAMPLE 10. Simplify:
12x5 y 7
4x3 y 2
Solution: We first express the fraction as a product of three fractions, the
latter two with a common base. In the first line of the following solution,
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
11
note that if you multiply numerators and denominators of the three separate
fractions, the product equals the original fraction on the left.
12 x5 y 7
12x5 y 7
=
·
·
4x3 y 2
4 x3 y 2
= 3x5−3 y 7−2
= 3x2 y 5
Break into a product of three fractions.
Simplify: 12/4=3. Then repeat the common
bases and subtract the exponents.
Simplify.
!
You Try It!
5n−4
x
x3−2n
Solution: Again, we repeat the base and subtract the exponents.
EXAMPLE 11. Simplify:
x5n−4
= x(5n−4)−(3−2n)
x3−2n
= x5n−4−3+2n
Repeat the base, subtract exponents.
Distribute the minus sign.
7n−7
=x
Simplify. Combine like terms.
!
Raising a Power to a Power
Suppose we have an exponential expression, raised to a second power, such as
(x2 )3 . The second exponent tells us to write x2 as a factor three times:
(x2 )3 = x2 · x2 · x2
= x6
Write x2 as a factor three times.
Repeat the base, add the exponents.
Note how we added 2 + 2 + 2 to get 6. However, a much faster way to add
“three twos” is to multiply: 3 · 2 = 6. Thus, when raising a “power to a second
power,” repeat the base and multiply the exponents, as follows:
(x2 )3 = x2·3
= x6
The preceding discussion gives rise to the following third law of exponents.
Raising a Power to a Power. When raising a power to a power, repeat the
base and multiply the exponents. In symbols:
(am )n = amn
Note that juxtaposing two variables, as in mn, means “m times n.”
12
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
You Try It!
EXAMPLE 12. Simplify each of the following expressions:
(a) (z 3 )5
(b) (73 )4
(c) [(x − y)3 ]6
Solution: In each example we are raising a power to a power. Hence, in each
case, we repeat the base and multiply the exponents.
(a) (z 3 )5 = z 3·5
(b) (73 )4 = 73·4
(c) [(x − y)3 ]6 = (x − y)3·6
= z 15
= 712
= (x − y)18
With a little practice, each of the examples can be simplified mentally. Repeat
the base and multiply the exponents in your head: (z 3 )5 = z 15 , (73 )4 = 712 ,
and [(x − y)3 ]6 = (x − y)18
!
You Try It!
EXAMPLE 13. Simplify:
(x2n−3 )4
Solution: Again, we repeat the base and multiply the exponents.
(x2n−3 )4 = x4(2n−3)
8n−12
=x
Repeat the base, multiply exponents.
Distribute the 4.
!
Raising a Product to a Power
We frequently have need to raise a product to a power, such as (xy)3 . Again,
remember the exponent is telling us to write xy as a factor three times, so:
(xy)3 = (xy)(xy)(xy)
= xyxyxy
= xxxyyy
= x3 y 3
Write xy as a factor three times.
The associative property allows us to
group as we please.
The commutative property allows us to
change the order as we please.
Invoke the exponent definition:
xxx = x3 and yyy = y 3 .
However, it is much simpler to note that when you raise a product to a power,
you raise each factor to that power. In symbols:
(xy)3 = x3 y 3
The preceding discussion leads us to a fourth law of exponents.
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
13
Raising a Product to a Power. To raise a product to a power, raise each
factor to that power. In symbols:
(ab)n = an bn
You Try It!
EXAMPLE 14. Simplify each of the following expressions:
(a) (yz)5
(b) (−2x)3
(c) (−3y)2
Solution: In each example we are raising a product to a power. Hence, in
each case, we raise each factor to that power.
(a) (yz)5 = y 5 z 5
(b) (−2x)3 = (−2)3 x3
(c) (−3y)2 = (−3)2 y 2
= −8x3
= 9y 2
With a little practice, each of the examples can be simplified mentally. Raise
each factor to the indicated power in your head: (yz)5 = y 5 z 5 , (−2x)3 = −8x3 ,
and (−3y)2 = 9y 2
!
When raising a product of three factors to a power, it is easy to show that we
should raise each factor to the indicated power. For example, (abc)3 = a3 b3 c3 .
In general, this is true regardless of the number of factors. When raising a
product to a power, raise each of the factors to the indicated power.
You Try It!
EXAMPLE 15. Simplify:
(2a3 b2 )3
Solution: Raise each factor to the third power, then simplify.
(2a3 b2 )3 = 23 (a3 )3 (b2 )3
9 6
= 8a b
Raise each factor to the third power.
Simplify: 23 = 8. In the remaining factors,
raising a power to a power requires that
we multiply the exponents.
!
You Try It!
EXAMPLE 16. Simplify:
(−2x2 y)2 (−3x3 y)
14
MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS
Solution: In the first grouped product, raise each factor to the second power.
(−2x2 y)2 (−3x3 y) = ((−2)2 (x2 )2 y 2 )(−3x3 y) Raise each factor in the first
grouped product to the second
power.
= (4x4 y 2 )(−3x3 y)
Simplify: (−2)2 = 4 and
(x2 )2 = x4 .
The associative and commutative property allows us to multiply all six factors
in the order that we please. Hence, we’ll multiply 4 and −3, then x4 and x3 ,
and y 2 and y, in that order. In this case, we repeat the base and add the
exponents.
= −12x7 y 3
Simplify: (4)(−3) = −12. Also,
x4 x3 = x7 and y 2 y = y 3 .
!
Raising a Quotient to a Power
Raising a quotient to a power is similar to raising a product to a power. For
example, raising (x/y)3 requires that we write x/y as a factor three times.
% &3
x x x
x
= · ·
y
y y y
x·x·x
=
y·y·y
x3
= 3
y
However, it is much simpler to realize that when you raise a quotient to a
power, you raise both numerator and denominator to that power. In symbols:
% &3
x
x3
= 3
y
y
This leads to the fifth and final law of exponents.
Raising a Quotient to a Power. To raise a quotient to a power, raise both
numerator and denominator to that power. In symbols:
' a (n
b
=
an
bn
4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF
EXPONENTS
15
You Try It!
EXAMPLE 17. Simplify each of the following expressions:
%
&4
% &2
' x (3
2
2
(b)
(c) −
(a)
3
3
y
Solution: In each example we are raising a quotient to a power. Hence, in
each case, we raise both numerator and denominator to that power.
(a)
% &2
22
2
= 2
3
3
4
=
9
(b)
' x (3
3
x3
33
x3
=
27
=
(c)
&4
%
24
2
−
= 4
y
y
16
= 4
y
Note that in example (c), raising a negative fraction to an even power produces
a postive result. With a little practice, each of the examples can be simplified
mentally. Raise numerator and denominator to the indicated power in your
head: (2/3)2 = 4/9, (x/3)3 = x3 /27, and (−2/y)4 = 16/y 4
!
You Try It!
EXAMPLE 18. Simplify:
%
&2
2x2
y3
Solution: Raise both numerator and denominator to the second power, then
simplify.
%
2x2
y3
&2
=
(2x2 )2
(y 3 )2
Raise numerator and denominator to the
second power.
In the numerator, we need to raise each factor of the product to the second
power. Then we need to remind ourselves that when we raise a power to a
power, we multiply the exponents.
=
22 (x2 )2
(y 3 )2
=
4x4
y6
Raise each factor in the numerator
to the second power
Simplify: 22 = 4, (x2 )2 = x4 ,
and (y 3 )2 = y 6 .
!