Algebra 2/Pre-Calculus
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The Factor and Remainder Theorems (Day 8, Polynomials)
In this handout, we will explore the factor theorem and the remainder theorem. (Don’t worry if
you don’t know what those theorems are – You’ll learn them as part of this problem set.)
1. In this problem, we will investigate different ways of representing the solution to a long
division problem.
a. Use long division to find the value of
of your answer.
60
7
. Make sure you include your remainder as part
b. The division in the last problem can be summarized in either of the following ways:
First way:
60
7
 8  74
Second way: 60  8  7  4
x4  2x3  4x2  9x  3
. Summarize your result in two
x 2  3x  2
different ways, just as we did with the division 607 .
Now perform the division
c. Here are the two ways of summarizing the division from part b:
First way:
x4  2x3  4x2  9x  3
2x  3
 x2  x  3  2
2
x  3x  2
x  3x  2
Second way: x 4  2 x 3  4 x 2  9 x  3  ( x 2  3x  2)( x 2  x  3)  2 x  3
Now we will generalize this idea. Suppose a division can be written as
p( x )
r( x)
 q( x ) 
. How would you write this division the other way?
d ( x)
d ( x)
2. In general, there are two ways to describe a polynomial division:
First way:
p( x )
r( x)
 q( x ) 
d ( x)
d ( x)
Second way: p( x )  d ( x )  q( x )  r ( x )
Our goal in this problem is to understand properties for the remainder r ( x ) .
a. Consider the long division from problem 1:
x 4  2 x 3  4 x 2  9 x  3  ( x 2  3x  2)( x 2  x  3)  2 x  3
Which functions are p ( x ) , d ( x ) , q(x ) , and r ( x ) in this example?
b. In general, how does the degree of the remainder r ( x ) relate to the degree of the
divisor d ( x ) ? Could the degree of the remainder be more than the degree of the
divisor? Could the degree of the remainder be equation to the degree of the divisor?
Explain. Hint: Think about example of long division that we’ve already seen.
c. Explain why the degree of the r ( x ) must be less than the degree of d ( x ) . (You may
assume that the degree of d ( x ) is at least 1.) Note: If you already explained this in part
b, you can move on to the next problem.
d. (Optional) In part c, we assumed that the degree of d ( x ) was at least one. Why do we
have to make this assumption?
3. This problem will lead us to an important conclusion involving the remainder.
x 3  9 x 2  26 x  7
a. Perform the division
. What is the remainder?
x3
b. Suppose p( x )  x 3  9 x 2  26 x  7 . Find the value of p(3) .
c. Perform the division
x 4  7 x 3  13x 2  21x  19
. What is the remainder?
x 5
d. Suppose p( x )  x 4  7 x 3  13x 2  21x  19 . Find the value of p(5) .
e. Look back parts a-d. Notice anything surprising?
f. Suppose we are performing the division
we say about the remainder?
p( x )
where p ( x ) is a polynomial. What can
xk
The idea that you explored in problem 3 is called the remainder theorem. Here’s the theorem:
Remainder Theorem Suppose p ( x ) is a polynomial. Then the remainder of
p( x )
is
xk
given by p (k ) .
4. Optional Exploration The goal of this problem is to explain the remainder theorem.
p( x )
. What will the degree of the remainder be?
xk
Hint: Think about the conclusion from problem 2.
a. Suppose we are doing the division
All too often, they just memorize the theorem!
b. Since the degree of the remainder is less than the degree of the divisor, we can conclude
p( x )
 q( x )  r where r is constant. How can
that the remainder has degree 0. Hence,
xk
we write this equation in the other form?
c. We can write this equation as p( x )  q( x )  ( x  k )  r . What do we get if we use this
equation to find the value of p (k ) ?
d. You should have found that p( k )  r . Explain why this completes our proof of the
remainder theorem.
5. In this problem, we explore some uses of the remainder theorem.
2 x 5  7 x 4  11x 3  24 x 2  38 x  49
a. What is the remainder for the division
?
x 1
Note: Don’t actually do this division. Use the remainder theorem!
b. You should have found that the remainder was -29. (Let
p( x )  2 x 5  7 x 4  11x 3  24 x 2  38 x  49 . Then p(1) = -29, so the remainder is -29.)
What is the remainder for the division
x 7  119
?
x2
x 3 + 4 x 2 + ax - 38
results in a remainder of 4. What is the value of a?
x -3
Hint: Remainder theorem again!
c. The division
d. Here’s the way to go about the last problem: Start with the equation
33 + 4(3)2 + 3a - 38 = 4. (Why this equation?) Then solve to find a = -7.
The division
x 4  ax 3  22 x 2  78 x  23
results in a remainder of -3. What is the value
x3
of a?
e. (Optional Challenge Problem) The division
x 3  ax 2  bx  17
results in a remainder
x2
x 3  ax 2  bx  17
of 5. The division
results in a remainder of -13. What are the values
x 1
of a and b? Hint: Set up and solve a system of equations.
6. In this problem, we will continue to explore the consequences of the remainder theorem.
x 3  7 x 2  19 x  28
a. What is the remainder for the division
? (Use the remainder
x4
theorem.)
b. You should have found that the remainder was 0. What can we say about x  4 ?
c. Since the remainder is 0, we can say that x  4 is a factor of x 3  7 x 2  19 x  28 .
x 3  7 x 2  19 x  28
Actually do the division
. What is the quotient?
x4
x 3  7 x 2  19 x  28
 x 2  3x  7 . This means
x4
x 3  7 x 2  19 x  28  ( x  4)( x 2  3x  7) . (Notice that x  4 really is a factor of
d. You should have found that
x 5  10000
. Is there any way to tell if
x  10
x  10 is a factor of x 5  10000 without doing long division?
x 3  7 x 2  19 x  28 .) Now consider the division
p( x )
. Is there any way to tell if x  k is a factor of p ( x )
xk
without doing long division?
e. Consider the division
The idea that you explored in problem 6 is called the factor theorem. Here’s the theorem:
Factor Theorem Suppose p ( x ) is a polynomial. If p( k )  0 then x  k is a factor of
p( x ) .
7. In this problem, we will explore some uses of the factor theorem.
a. Is x  4 a factor of x 4  6 x 3  13x 2  27 x  28 ? How can you tell?
b. Is x  1 a factor of x 6  x 5  3x 2  4 x  10 ? How can you tell?
c. What value of a would make x  5 a factor of x 3  ax 2  18 x  15 ?
d. What value of a would make x  1 a factor of x 5  x 4  ax 3  2 x 2  x ?
e. (Optional Challenge) What values of a and b would make x  2 and x  4 factors of
x 4  ax 3  21x 2  22 x  b ?