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![Page 1: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/1.jpg)
Objectives:1. To divide polynomials using long and synthetic
division2. To apply the Factor and Remainder Theorems to
find real zeros of polynomial functions
![Page 2: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/2.jpg)
As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook.
Quotient Remainder
Dividend Divisor
Divides Evenly
Factor
![Page 3: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/3.jpg)
Use long division to divide 5 into 3462.
5 34626
30-
46
9
45-
12
2
10-
2
![Page 4: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/4.jpg)
Use long division to divide 5 into 3462.
5 34626
30-
46
9
45-
12
2
10-
2
Divisor Dividend
Quotient
Remainder
![Page 5: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/5.jpg)
Use long division to divide 5 into 3462.
3462 2692
5 5
Dividend
Divisor
Quotient
Remainder
Divisor
![Page 6: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/6.jpg)
If you are lucky enough to get a remainder of zero when dividing, then the divisor divides evenlydivides evenly into the dividend.
This means that the divisor is a factorfactor of the dividend.
For example, when dividing 3 into 192, the remainder is 0. Therefore, 3 is a factor of 192.
![Page 7: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/7.jpg)
Dividing polynomials works just like long division. In fact, it is called long divisionlong division!
Before you start dividing:
1. Make sure the divisor and dividend are in standard form (highest to lowest powers).
2. If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder.
32 3x x 3 22 0 3x x x
![Page 8: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/8.jpg)
Divide x + 1 into x2 + 3x + 5
Line up the first term of the quotient with the term of the dividend with the same degree.
21 3 5x x x
How many times does x go into x2?x
Multiply x by x + 1
2 x x-
2x-
5
2
Multiply 2 by x + 1
2 2x - -
3
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Divide x + 1 into x2 + 3x + 5
21 3 5x x x x
2 x x-
2x-
5
2
2 2x - -
3Divisor
Dividend
Quotient
Remainder
![Page 10: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/10.jpg)
Divide x + 1 into x2 + 3x + 5
2 3 5 32
1 1
x xx
x x
Divisor
Dividend
Quotient
Remainder
Divisor
![Page 11: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/11.jpg)
Divide 6x3 – 16x2 + 17x – 6 by 3x – 2
![Page 12: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/12.jpg)
Use long division to divide x4 – 10x2 + 2x + 3 by x – 3
![Page 13: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/13.jpg)
When your divisor is of the form x - k, where k is a constant, then you can perform the division quicker and easier using just the coefficients of the dividend.
This is called fake division. I mean, synthetic divisionsynthetic division.
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Synthetic Division Synthetic Division (of a Cubic Polynomial)To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.k a b c d
a
ka
= Add terms
= Multiply by k
Coefficients of Quotient (in decreasing order)
Remainder
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Synthetic DivisionSynthetic Division (of a Cubic Polynomial)To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.
Important Note: Important Note: You are always adding columns using synthetic division, whereas you subtracted columns in long division.
k a b c d
a
ka
= Add terms
= Multiply by k
![Page 16: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/16.jpg)
Synthetic Division Synthetic Division (of a Cubic Polynomial)To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.
Important Note: Important Note: k can be positive or negative. If you divide by x + 2, then k = -2 because x + 2 = x – (-2).
k a b c d
a
ka
= Add terms
= Multiply by k
![Page 17: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/17.jpg)
Synthetic Division Synthetic Division (of a Cubic Polynomial)To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.
Important Note: Important Note: Add a coefficient of zero for any missing terms!
k a b c d
a
ka
= Add terms
= Multiply by k
![Page 18: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/18.jpg)
Use synthetic division to divide x4 – 10x2 + 2x + 3 by x – 3
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Evaluate f (3) for f (x) = x4 – 10x2 + 2x + 3.
![Page 20: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/20.jpg)
If a polynomial f (x) is divided by x – k, the remainder is r = f (k).
This means that you could use synthetic division to evaluate f (5) or f (-2). Your answer will be the remainder.
![Page 21: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/21.jpg)
Divide 2x3 + 9x2 + 4x + 5 by x + 3 using synthetic division.
![Page 22: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/22.jpg)
Use synthetic division to divide f(x) = 2x3 – 11x2 + 3x + 36 by x – 3.
Since the remainder is zero when dividing f(x) by x – 3, we can write:
This means that x – 3 is a factorfactor of f(x).
2( )2 5 12,
3
f xx x
x
2 so ( ) ( 3)(2 5 12)f x x x x
![Page 23: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/23.jpg)
A polynomial f(x) has a factor x – k if and only if f(k) = 0.
This theorem can be used to help factor/solve a polynomial function if you already know one of the factors.
![Page 24: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/24.jpg)
Factor f(x) = 2x3 – 11x2 + 3x + 36 given that x – 3 is one factor of f(x). Then find the zeros of f(x).
![Page 25: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/25.jpg)
Given that x – 4 is a factor of x3 – 6x2 + 5x + 12, rewrite x3 – 6x2 + 5x + 12 as a product of two polynomials.
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Find the other zeros of f(x) = 10x3 – 81x2 + 71x + 42 given that f(7) = 0.
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Rational Zero Test: we use this to find the rational zeros for a polynomial f(x). It says that if f(x) is a polynomial of the form:
1 2 11 2 1 0( ) n n
n nf x a x a x a x a x a
Then the rational zeros of f(x) will be of the form:p
qRational zero =
Possible rational zeros = factors of the constant term___factors of the leading coefficient
Where p = factor of the constant &
q = factor of leading coefficient
•Keep in mind that a polynomial can have rational zeros, irrational zeros and complex zeros.
p
q
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Ex 1: Find all of the possible rational zeros of f(x)
4 3 2( ) 3 6f x x x x x
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Ex 2: Find the rational zeros of: 4 3 2( ) 3 6f x x x x x
Let’s start by listing all of the possible rational zeros, then we will use synthetic division to test out the zeros:
1. Start with a list of factors of -6 (the constant term): p =
2. Next create a list of factors of 1 (leading coefficient): q =
3. Now list your possible rational zeros: p/q =
Testing all of those possibilities could take a while so let’s use the graph of f(x) to locate good possibilities for zeros.
Use your trace button!
![Page 30: Objectives: 1. To divide polynomials using long and synthetic division 2. To apply the Factor and Remainder Theorems to find real zeros of polynomial functions.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ed55503460f94be5d63/html5/thumbnails/30.jpg)
4 3 2( ) 3 6f x x x x x
Ex 2 continued: Find all of the rational zeros of the function
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Ex 3: Find all the real zeros of :
p = Factors of 3:
q = Factors of 2:
Candidates for rational zeros: p/q =
Let’s look at the graph: Which looks worth trying?
Now use synthetic division to test them out.
3 22 3 8 3 0x x x
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Homework
Dividing Polynomials Worksheet
Page 127-128
36,38, 49-59 odd