2.3: polynomial division
Post on 06-Jan-2016
120 Views
Preview:
DESCRIPTION
TRANSCRIPT
Objectives:1. To divide polynomials using long and synthetic
division2. To apply the Factor and Remainder Theorems to
find real zeros of polynomial functions
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
Use long division to divide 5 into 3462.
5 34626
30-
46
9
45-
12
2
10-
2
Divisor Dividend
Quotient
Remainder
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.
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
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
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
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.
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
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
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
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
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.
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
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.
Factor f(x) = 2x3 – 11x2 + 3x + 36 given that x – 3 is one factor of f(x). Then find the zeros of f(x).
Given that x – 4 is a factor of x3 – 6x2 + 5x + 12, rewrite x3 – 6x2 + 5x + 12 as a product of two polynomials.
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
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!
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
top related