Algebra 2 B Notes: 5.5-5.8
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5.5 Theorems About Roots of Polynomial Equations
Our goal:
Find all the roots (__________________, ___________________)
real and imaginary for polynomial equations.
In Lesson 5.3 we found solutions by factoring, but sometimes this method is very difficult.
Today:
Discuss some theorems about polynomials that are true to help us with lesson 5.6 where we will
be solving polynomial equations again using a different strategy.
A: The Rational Root Theorem
Rational number:
A number that can be written in the form
P
, where P and Q are both integers.
Q
Examples:
Rational numbers:
Irrational numbers:
One way to find a solution is to guess and check.
Example:
Find a rational zero for x 2  3x  10  0
Example:
Find a rational zero for: 15 x3  32 x 2  3x  2  0
We could still use guess and check, but we need some guidance/strategy as to what kinds of numbers to start
guessing. We are going to use the “Rational Root Theorem” to make a list of POSSIBILITIES.
Algebra 2 B Notes: 5.5-5.8
Translation:
If
2
P  a factor of the constant term
P
is a rational root of our polynomial, then
Q  a factor of the leading coefficient
Q
To make a list of possible rational
roots:
Example 1: Use the rational root theorem to list all possible
rational roots for each equation.
1. Write down all possible factors of
your constant term.
a.
15 x3  32 x 2  3x  2  0
b.
2 x3  x 2  7 x  6  0
2. Write down all possible factors of
your leading coefficient.
3. Systematically write down all
numbers from part (1.) over all
numbers from part (2.)
4. Put a  in front of your list.
c.
x3  27 x  10  0
Algebra 2 B Notes: 5.5-5.8
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B: The Conjugate Root Theorem
Remember that roots can be real or imaginary, rational or irrational.
Another useful tool in finding all the zeros of a polynomial function is the “Conjugate Root Theorem” for
irrational and complex(imaginary) numbers.
Translation:
Irrational roots will ALWAYS come in _____________ of CONJUGATES!!!
This means if:
6  3 is a solution, then so is _______________!
This means if:
 5 is a solution, then so is _________!
a b
and a  b
Complex roots (imaginary roots) will ALWAYS come in ____________ of CONJUGATES!!! a  bi and
This means if:
This means if:
5  7i is a solution, then so is _______________!
3i is a solution, then so is _______ !
Example 2:
A polynomial P(x) with rational coefficients has the given roots. Find two additional roots of P(x).
a.
2
, 3  7,
5
 4i
b.
5, 3, 4  7i,  17
a  bi
Algebra 2 B Notes: 5.5-5.8
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Example 3:
Write the polynomial with rational coefficients so that P(x) = 0 and has the given roots.
Recall: To write a
polynomial
a)
7, 10
b)
5i
d)
1,  2, -3i
1. Write out the factors
a  (x – a )
2. Multiply carefully
using FOIL
techniques
c)
e)
5 , 4i
3  2i
f) 9  2
Algebra 2 B Notes: 5.5-5.8
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5.6 The Fundamental Theorem of Algebra
A: How many roots should there be?
The _______________ of a polynomial tells you how many roots the polynomial has total.
These roots may be real, or complex (imaginary).
A real zero (rational or irrational) shows up as an ________________________ on the graph of the equation.
Example 1:
How many roots does the equation have?
12 x5  3x 2  17 x10  0
B: How can you tell if a number is actually a root?
Given a polynomial P(x), then x = a is a root if P(a) = 0.
To check:
 Solve using factoring techniques like section 5.3
 Guess and check randomly like section 5.5
 SYSTEMATICALLY guess and check using synthetic substitution.
Algebra 2 B Notes: 5.5-5.8
Example 2:
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Find all the roots, real & complex (imaginary) of the given polynomial.
Make it happen:
a)
y  x3  2 x 2  11x  12
1. Determine how many roots
the polynomial must have.
2. Make a list of POSSIBLE
rational zeros (lesson 5.5).
3. Use synthetic substitution
using your numbers from part
(2.) until you get a remainder
of zero. This number is a root.
4. If your result (the quotient) is
degree 2, solve using factoring,
quadratic formula, etc. (lessons
4.4-4.7).
If your result is not degree 2,
repeat step 3 with your
quotient.
5. List out all your answers.
b)
y  x3  4 x 2  8
c)
y  x3  4 x 2  7 x  28
Algebra 2 B Notes: 5.5-5.8
d)
y  x 4  7 x 2  12
f) y  x5  7 x 4  7 x3  17 x 2  8x  10
7
e) y  x 4  2 x3  x 2  4 x  6
Algebra 2 B Notes: 5.5-5.8
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5.7 The Binomial Theorem
Pascal’s Triangle:
To make a Pascal’s Triangle:
A: Expanding Binomials
You can expand any binomial using the distributive property:
After a few of these expansions of  a  b  , a pattern occurs:
n
Notice:
When expanding
a  b
n
Powers of a:
Powers of b:
Sum of exponents:
Coefficients come from:
Example: Expand
( a  b) 6
Algebra 2 B Notes: 5.5-5.8
Example 1:
Use the Binomial Theorem to Expand the following.
a.)
 x  5
b)
 3x  5
c)
 2x  3y 
4
3
5
9
Algebra 2 B Notes: 5.5-5.8
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B: Specific Terms in an expansion
If you only want a specific term for an expansion, there’s a pattern for that too…
Specific Term of a Binomial Expansion of (a  b)n =
n! m k
a b
m !k !
Example 2:
Find the indicated term of each binomial expansion
a) The 3rd term of  x  5 
b) The 2nd term of (2 y  3x) 4
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5.8 Polynomial Models in the Real World
Sometimes we can predict what’s going to happen by analyzing behavior.
Consider this data:
Which “model” (equation type) seems to match the data points best?
Linear
Quadratic
Cubic
Your graphing calculator has three different “regression” models to help predict behavior.
LinReg
QuadReg
CubicReg
y  mx  b
y  ax  bx  c
y  ax3  bx 2  cx  d
m=
b=
r2=
a=
b=
a=
c=
r2 =
r2 =
2
b=
c=
d=
The closer r2
is to 1 , the
better the
equation
“fits” the
data.
Algebra 2 B Notes: 5.5-5.8
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Find a model for each set of data points
Example 1:
a) The graphic shows the average milk production.
Find a LINEAR model for the data.
Estimate the amount of milk production in 1995.
b) The table shows cheese production per person for the selected years.
Find a CUBIC model for the data.
Predict the amount of cheese production in 2012.
c) Find a QUADRATIC model for the Cheese Graph in part (b).
Predict the amount of cheese production in 2012.
d) Which model from (b.) and (c.) is the best? How can you tell?