The
ability
to
concentrate
depends on adequate sleep,
decent nutrition,and the phys-
ical
well-being
that
comes
with exercise.

Solution
a)
y
9
5
y
4
Use the de
fi
nition of division to check that
y
4
y
5
y
9
.
b)
4
Use the de
fi
nition of division to check that
3
b
7
12
b
2
.
c)
6
x
3
(2
x
9
)
Use the de
fi
nition of division to check that
2
x
9
6
x
3
.
d)
x
6
y
0
x
6
Use the de
fi
nition of division to check that
x
6
x
2
y
2
x
8
y
2
.
■
We showed more steps in Example 2 than are necessary. For division problems
like these you should try to write down only the quotient.
Dividing a Polynomial by a Monomial
We divided some simple polynomials by monomials in Chapter 1. For example,
(6
x
8)
3
x
4.
We use the distributive property to take one-half of 6
x
and one-half of 8 to get
3
x
4. So both 6
x
and
8 are divided by 2. To divide any polynomial by a mono-
mial, we divide each term of the polynomial by the monomial.
E X A M P L E 3
Dividing a polynomial by a monomial
Find the quotient for (
8
x
6
12
x
4
4
x
2
)
(4
x
2
).
Solution
2
x
4
3
x
2
1
The quotient is
2
x
4
3
x
2
1. We can check by multiplying.
4
x
2
(
2
x
4
3
x
2
1)
8
x
6
12
x
4
4
x
2
.
■
Because division by zero is unde
fi
ned, we will always assume that the divisor
is nonzero in any quotient involving variables. For example, the division in Exam-
ple 3 is valid only if 4
x
2
0, or
x
0.
4
x
2
4
x
2
12
x
4
4
x
2
8
x
6
4
x
2
8
x
6
12
x
4
4
x
2
4
x
2
1
2
6
x
8
2
y
2
y
2
x
8
x
2
x
8
y
2
x
2
y
2
6
x
9
x
6
3
x
6
3
x
6
6
x
3
2
x
9
12
b
7
b
5
4
b
5
4
b
5
1
b
7
2
b
2
b
7
12
3
12
b
2
3
b
7
y
9
y
5
232
(4-26)
Chapter 4
Polynomials and Exponents
h e l p f u l
h i n t
s t u d y
t i p
Play offensive math,not defen-
sive math.A student who says,
“Give me a question and I’ll see
if I can answer it,” is playing
defensive math. The student
is taking a passive approach
to learning. A
student
who
takes an active approach and
knows the usual questions
and answers for each topic is
playing offensive math.
Recall that the order of oper-
ations
gives
multiplication
and division an equal ranking
and says to do them in order
from left to right. So without
parentheses,
6
x
3
2
x
9
actually means
x
9
.
6
x
3
2

Dividing a Polynomial by a Binomial
Division of whole numbers is often done with a procedure called
long division.
For
example, 253 is divided by 7 as follows:
36
←
Quotient
Divisor
→
7 253
←
Dividend
21
43
42
1
←
Remainder
Note that 36
7
1
253. It is always true that
(quotient)(divisor)
(remainder)
dividend.
To divide a polynomial by a binomial, we perform the division like long divi-
sion of whole numbers. For example, to divide
x
2
3
x
10 by
x
2, we get the
fi
rst term of the quotient by dividing the
fi
rst term of
x
2 into the
fi
rst term of
x
2
3
x
10. So divide
x
2
by
x
to get
x
, then multiply and subtract as follows:
1
Divide:
x
x
2
x
x
2
Multiply:
x
2
x
2
3
x
10
x
2
2
x
x
(
x
2)
x
2
2
x
3
Subtract:
5
x
3
x
2
x
5
x
Now bring down
10 and continue the process. We get the second term of the quo-
tient (below) by dividing the
fi
rst term of
x
2 into the
fi
rst term of
5
x
10. So
divide
5
x
by
x
to get
5:
1
Divide:
x
5
5
x
x
5
2

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- Spring '14
- AlexanderFrenkel
- Calculus, Polynomials, Addition, Exponents, Coefficient