Write a post in which you discuss the differences and similarities between the Factor Theorem and the Remainder Theorem. Try to first restate each one in simple terms the way that you currently understand them. Then, explain why you think both of these theorems are used in mathematics (i.e., why couldn't we just have one or the other?). There isn't a "right answer" here, so try to express yourself to be understood. You should consider what the theorems say and how you would explain them to someone new to Algebra 2, though not necessarily new to Algebra. This post is where I have seen the most copying from the internet. I know what they all are and I've seen them all before. If I see anything from these or any other websites, I will automatically report you for plagiarism. I will not tolerate it. Just answer the question in your own words. Thank you.
Answers
Answer:
The remainder theorem tells us that for any polynomial
f
(
x
)
, if you divide it by the binomial
x
−
a
, the remainder is equal to the value of
f
(
a
)
.
The factor theorem tells us that if
a
is a zero of a polynomial
f
(
x
)
, then
(
x
−
a
)
is a factor of
f
(
x
)
, and vice-versa.
For example, let's consider the polynomial
f
(
x
)
=
x
2
−
2
x
+
1
Using the remainder theorem
We can plug in
3
into
f
(
x
)
.
f
(
3
)
=
3
2
−
2
(
3
)
+
1
f
(
3
)
=
9
−
6
+
1
f
(
3
)
=
4
Therefore, by the remainder theorem, the remainder when you divide
x
2
−
2
x
+
1
by
x
−
3
is
4
.
You can also apply this in reverse. Divide
x
2
−
2
x
+
1
by
x
−
3
, and the remainder you get is the value of
f
(
3
)
.
Using the factor theorem
The quadratic polynomial
f
(
x
)
=
x
2
−
2
x
+
1
equals
0
when
x
=
1
.
This tells us that
(
x
−
1
)
is a factor of
x
2
−
2
x
+
1
.
We can also apply the factor theorem in reverse:
We can factor
x
2
−
2
x
+
1
into
(
x
−
1
)
2
, therefore
1
is a zero of
f
(
x
)
.
Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.
Answer:
The remainder theorem tells us that for any polynomial
f
(
x
)
, if you divide it by the binomial
x
−
a
, the remainder is equal to the value of
f
(
a
)
.
The factor theorem tells us that if
a
is a zero of a polynomial
f
(
x
)
, then
(
x
−
a
)
is a factor of
f
(
x
)
, and vice-versa.
For example, let's consider the polynomial
f
(
x
)
=
x
2
−
2
x
+
1
Using the remainder theorem
We can plug in
3
into
f
(
x
)
.
f
(
3
)
=
3
2
−
2
(
3
)
+
1
f
(
3
)
=
9
−
6
+
1
f
(
3
)
=
4
Therefore, by the remainder theorem, the remainder when you divide
x
2
−
2
x
+
1
by
x
−
3
is
4
.
You can also apply this in reverse. Divide
x
2
−
2
x
+
1
by
x
−
3
, and the remainder you get is the value of
f
(
3
)
.
Step-by-step explanation:
The quadratic polynomial
f
(
x
)
=
x
2
−
2
x
+
1
equals
0
when
x
=
1
.
This tells us that
(
x
−
1
)
is a factor of
x
2
−
2
x
+
1
.
We can also apply the factor theorem in reverse:
We can factor
x
2
−
2
x
+
1
into
(
x
−
1
)
2
, therefore
1
is a zero of
f
(
x
)
.
Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.
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