Transcript Polynomial Long Division

Polynomial Long Division
Chapter 5.5
Operations on Polynomials
• In earlier lessons, you learned that polynomials have a structure that is
similar to that of numbers
• A number like 472 can be expanded as 4 × 102 + 7 × 10 + 2
• This has the same structure as the quadratic expression 4𝑥 2 + 7𝑥 + 2,
with x replacing 10
• This led us to see that we can perform operations on polynomials in
much the same way we perform operations on numbers
• You learned to add/subtract polynomials by combining like terms, and
to multiply polynomials by using the Distributive Property
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Operations on Polynomials
• You saw that we can even raise a polynomial to a power, such as
𝑥 + 3 2 by again applying the Distributive Property
• We were able to find an easy way to factor quadratic polynomials
(when possible), which is related to division
• However, we didn’t actually divide one polynomial by another
• Is it possible to use long division on polynomials the same way (or in a
similar way) to using long division with numbers?
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What is Division?
• To begin to answer this question, let’s first explore a related question:
does dividing one polynomial by another even have any meaning?
• First, let’s recall how we defined division near the beginning of the
course
• Suppose a, b, and c are real numbers, where neither a nor b is equal to
zero
• Then if 𝑎𝑏 = 𝑐 ⟺ 𝑎 = 𝑐 ÷ 𝑏 or 𝑏 = 𝑐 ÷ 𝑎
• We see this is true if we replace a, b, and c with 4, 3, and 12:
4 × 3 = 12 ⟺ 4 = 12 ÷ 3 or 3 = 12 ÷ 4
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What is Division?
• We can apply the same idea to polynomials:
𝑥 + 2 𝑥 + 5 = 𝑥 2 + 7𝑥 + 10 ⟺ 𝑥 + 2 = 𝑥 2 + 7𝑥 + 10 ÷ (𝑥 + 5)
• So we can say that polynomial division does have meaning
• Why should we even want to divide polynomials?
• Dividing polynomials will allow us to solve polynomial equations in
the same way we did before by factoring
• Before examining division of polynomials, let’s first understand why
long division works with numerals
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What is Division?
• The long division algorithm is based on the following
• You know that these are all true statements:
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•
•
•
4×0=0
4×1=4
4×2=8
4 × 3 = 12, and so on
• Is there any way we can use a product of 4 to represent all whole
numbers?
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What is Division?
• We can do so like this:
•
•
•
•
•
•
•
•
4×0+0=0
4×0+1=1
4×0+2=2
4×0+3=3
4×1+0=4
4×1+1=5
4×1+2=6
4 × 1 + 3 = 7, and so on
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What is Division?
• Consider the following equation
7 = 4𝑞 + 𝑟
• This is the division algorithm in disguise because, to find q and r w
would ask, “By what whole number q can 4 be multiplied to come as
close to 7 as possible without going over?”
• In this case the answer is 1, so 7 = 4 ⋅ 1 + 𝑟
• This means that r must be 3: 7 = 4 ⋅ 1 + 3
• Note that 1 is the quotient (or the “answer” when dividing) and 3 is the
remainder
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What is Division?
• Now let’s take a look at division of a three-digit number by 4 (one that
divides evenly) and compare it to the division algorithm
• What is 252 ÷ 4?
• This is the same as asking, by what whole number can you multiply 4
to get a close to 252 as possible without going over?
• As an equation this is the solution (for q and r) to
252 = 4𝑞 + 𝑟
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What is Division?
252 = 4𝑞 + 𝑟
• We can find this in two separate steps
• In the first step, we treat the 250 of 252 as though it were 25
25 = 4𝑞1 + 𝑟1
• By what whole number 𝑞1 can 4 be multiplied to get as close as
possible to 25 without going over?
25 = 4 ⋅ 6 + 𝑟1
• Since 4 ⋅ 6 = 24, then 𝑟1 = 1
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What is Division?
• We now have 25 = 4 ⋅ 6 + 1
• But remember that we treated 250 as though it were 25
• We can multiply both sides of the equation by 10
25 ⋅ 10 = 4 ⋅ 6 ⋅ 10 + 1 ⋅ 10
250 = 4 ⋅ 60 + 10
• Now, we ignored the 2, so let’s add it back
250 + 2 = 4 ⋅ 60 + 10 + 2
252 = 4 ⋅ 60 + 12
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What is Division?
252 = 4 ⋅ 60 + 12
• Next, we perform the same procedure on 12 with the goal of
expressing it as a multiple of 4
• We have 12 = 4𝑞2 + 𝑟2
• Since 12 is already a multiple of 4 our equation becomes
12 = 4 ⋅ 3 + 0
• Now if we replace 12 in the above equation with 4 ⋅ 3 + 0
252 = 4 ⋅ 60 + 4 ⋅ 3 + 0
252 = 4 60 + 3 + 0 = 4 ⋅ 63 + 0
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What is Division?
252 = 4 ⋅ 63 + 0
• Now let’s go through the division algorithm to compare the steps
4 )252
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What is Division?
• In the previous example, 4 divides into 252 evenly
• How does this procedure compare using a number for which 4 does
not divide evenly?
227 ÷ 4
• We will treat 220 as though it were 22
22 = 4𝑞1 + 𝑟1
22 = 4 ⋅ 5 + 𝑟1
22 = 4 ⋅ 5 + 2
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What is Division?
22 = 4 ⋅ 5 + 2
• But 22 is really 220, so multiply both sides by 10
22 ⋅ 10 = 4 ⋅ 5 ⋅ 10 + 2 ⋅ 10
220 = 4 ⋅ 50 + 20
• Now add 7 to both sides of the equation
220 + 7 = 4 ⋅ 50 + 20 + 7
227 = 4 ⋅ 50 + 27
• We must express 27 using a multiple of 4, so we perform the same
steps
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What is Division?
27 = 4𝑞2 + 𝑟2
27 = 4 ⋅ 6 + 𝑟2
27 = 4 ⋅ 6 + 3
• Replace 27 with 4 ⋅ 6 + 3
227 = 4 ⋅ 50 + 4 ⋅ 6 + 3
• Factor out the common 4 from the two middle terms
227 = 4 50 + 6 + 3
227 = 4 ⋅ 56 + 3
We can divide the equation by 4 to get
227
4
3
4
= 56 + =
3
56
4
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What is Division?
• Now let’s carry out the long division algorithm by hand in order to
compare the steps
4 )227
• Note that in the equation 227 = 4𝑞 + 𝑟, the variable q is the quotient
(or what we often think of as the “answer”) and the variable r is the
remainder
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Polynomial Long Division Examples
• To see how this algorithm works using polynomials, let’s first look at
an example with remainder 0
• You can multiply (𝑥 + 4)(𝑥 − 5) to get 𝑥 2 − 𝑥 − 20
• Let’s assume that we want to divide 𝑥 2 − 𝑥 − 20 ÷ (𝑥 + 4) and that
we don’t know the answer
• We want to find 𝑞(𝑥) and 𝑟(𝑥) such that
𝑥 2 − 𝑥 − 20 = 𝑥 + 4 ⋅ 𝑞 𝑥 + 𝑟(𝑥)
• We use 𝑞(𝑥) and 𝑟(𝑥) instead of just q and r because these may
include x
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Polynomial Long Division Examples
𝑥 + 4) 𝑥 2 −𝑥 − 20
• So 𝑞 𝑥 = 𝑥 − 5 and 𝑟 𝑥 = 0 and the result is
𝑥 2 − 𝑥 − 20 = (𝑥 + 4)(𝑥 − 5)
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Polynomial Long Division Examples
• Next, let’s see how to carry out the algorithm that ends up with a
remainder other than zero
• Divide 𝑥 2 + 5𝑥 − 20 ÷ (𝑥 + 4)
• We must find 𝑞(𝑥) and 𝑟(𝑥) such that
𝑥 2 + 5𝑥 − 20 = 𝑥 + 4 ⋅ 𝑞 𝑥 + 𝑟(𝑥)
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Polynomial Long Division Examples
𝑥 + 4) 𝑥 2 +5𝑥 − 20
• So 𝑞 𝑥 = 𝑥 + 1 and 𝑟 𝑥 = −24 and the result is
𝑥 2 + 5𝑥 − 20 = 𝑥 + 4 𝑥 + 1 − 24
• This can also be written as
𝑥 2 + 5𝑥 − 20
27
=𝑥+1−
𝑥+4
𝑥+4
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Polynomial Long Division Examples
• In this next example, you will see how to insert 0 for missing terms
in order to properly align like terms
• Divide 3𝑥 4 − 5𝑥 3 + 4𝑥 − 6 ÷ (𝑥 2 − 3𝑥 + 5)
• Note that the 𝑥 2 term is missing from the dividend (the
polynomial being divided)
• So we write the division as
3𝑥 4 − 5𝑥 3 + 0 ⋅ 𝑥 2 + 4𝑥 − 6 ÷ 𝑥 2 − 3𝑥 + 5
• As we go through the steps of the algorithm, you will see how this
is helps you to proceed correctly
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Polynomial Long Division Examples
𝑥 2 − 3𝑥 + 5) 3𝑥 4 − 5𝑥 3 + 0 ⋅ 𝑥 2 − 3𝑥 + 5
• So 𝑞 𝑥 = 3𝑥 2 + 4𝑥 − 3 and 𝑟 𝑥 = −25𝑥 + 9 and the result is
3𝑥 4 − 5𝑥 3 − 3𝑥 + 5 = 𝑥 2 − 3𝑥 + 5 3𝑥 2 + 4𝑥 − 3 − 25𝑥 + 9
• This can also be written as
3𝑥 3 − 5𝑥 3 − 3𝑥 + 5
−25𝑥 + 9
2
= 3𝑥 + 4𝑥 − 3 + 2
2
𝑥 − 3𝑥 + 5
𝑥 − 3𝑥 + 5
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Guided Practice
Divide using polynomial long division.
a) 2𝑥 4 + 𝑥 3 + 𝑥 − 1 ÷ 𝑥 2 + 2𝑥 − 1
b)
c)
𝑥 3 − 𝑥 2 + 4𝑥 − 10 ÷ 𝑥 + 2
2𝑥 2 −9
𝑥−5
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Exercise 5.5
• Handout
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