Download Section 2.5

Section 2.5 – 2.7
DIVIDING POLYNOMIALS
REMAINDER AND FACTOR THEOREMS
FINDING ZEROS FOR POLYNOMIALS
Do-Now: Homework Quiz
 Factor the following expression.
 12x5 – 27x
 Solve the following equation.
 x3 + 6x2 - 40x = 0
5th grade challenge
 Try doing the following division problem without
using a calculator.
 8│45379
 How do you write the remainder in a division
problem?
Polynomial Long Division
 Divide f(x) = 2x4 + x3 + x – 1 by x2 + 2x – 1.
 Set your problem up like a long division problem.
 You must include a 0x2 term.
Polynomial Long Division
 Divide f(x) = 3x3 + 17x2 + 21x – 11 by x + 3.
 Set your problem up like a long division problem.
Long division vs. synthetic division
 Find the value of the previous function when x = -3
using synthetic substitution.
 Why did I choose x = -3?
 What do you notice about the value of the function?
 What do you notice about the other numbers in the
bottom row?
Remainder Theorem
 If a polynomial f(x) is divided by x – k, then the
remainder is equal to the value of the function
evaluated at x = k.
Additional example
 Divide using long and synthetic division.
 f(x) = 2x3 + 9x2 + 14x + 5 divided by x – 3.
Additional examples
 Perform the following division problems using
synthetic substitution.
 2x4 – 8x3 + 3x – 7 by x – 5
 Make sure you put a zero in place of the missing term.
 x3 + x2 – 16x – 16 by x – 4.
 What is the remainder?
 What does this tell you about the number 4 and the
expression x – 4?
Factor Theorem
 A polynomial f(x) has a factor x – k if
and only if f(k) = 0.
 This
occurs when the remainder is 0 if f(x) is
divided by x – k.
 In other words, x – k is a factor because f(x)
is evenly divisible by x – k.
Example
 Factor x2 + 3x – 18.
 Answer: (x + 6)(x – 3)
 Since (x + 6)(x – 3) = x2 + 3x – 18….
 we can do the inverse operation and come up with…
𝑥2+3𝑥 −18

𝑥 −3

=𝑥+6
Confirm this result using synthetic division.
 Now that we can divide polynomials, if we know one
factor, we can use it to find other factors.
Factoring given a factor
 Factor the polynomial: 2x3 – 11x2 + 3x + 36
 What method did you try. Why didn’t it work?
 What if I told you that one of the factors is x – 3?
How would that allow you to factor the polynomial?
 Use synthetic division to divide by the given factor.
Then see if you can factor the expression that results
from the division.
Additional Example
 Factor x3 + 9x2 + 23x + 15 if x + 5 is one of the
factors.
Using factoring to solve equations.
 Remember: If x – k is a factor of a polynomial, then





k is a zero of the polynomial
k is an x intercept of the graph of the polynomial
k is a root of the polynomial
if you substitute k into the equation the value of the equation is zero.
when the equation is set equal to zero, k is a solution.
 Example: If x – 4 is a factor, then 4 is a zero.
 Example: If x + 8 is a factor, then the graph has an x-
intercept at -8.
 Example: If -5 is a zero of a polynomial, then x + 5 is a
factor.
Example
 Solve the equation x3 – 7x2 + 2x + 40 = 0.
 One of the zeros is 5. Find the other two.
Additional Example
 Solve: 3x3 + 4x2 – 35x – 12 = 0.
 Hint: one of the solutions is x = 3.
EXAMPLE 6
Use a polynomial model
BUSINESS
The profit P (in millions
of dollars) for a shoe
manufacturer can be
modeled by
P = – 21x3 + 46x
where x is the number
of shoes produced
(in millions). The company
now produces 1 million shoes and makes a profit of
$25,000,000, but would like to cut back production.
What lesser number of shoes could the company
produce and still make the same profit?
EXAMPLE 6
Use a polynomial model
SOLUTION
Substitute 25 for P in P = – 21x3 + 46x.
25 = – 21x3 + 46x
0 = 21x3 – 46x + 25
Write in standard form.
You know that x = 1 is one solution of the equation.
This implies that x – 1 is a factor of 21x3 – 46x + 25.
Use synthetic division to find the other factors.
1
21
0
21
21
– 46
25
21 –25
21 – 25
0
EXAMPLE 6
Use a polynomial model
So, (x – 1)(21x2 + 21x – 25) = 0. Use the quadratic formula
to find that x ≈ 0.7 is the other positive solution.
ANSWER
The company could still make the same profit
producing about 700,000 shoes.
Daily Homework Quiz
4. One of the costs to print a novel can be modeled
by C = x3 – 10x2 + 28x, where x is the number of
novels printed in thousands. The company now
prints 5000 novels at a cost of $15,000. What
other numbers of novels would cost about the
same amount?
ANSWER
About 4300 or about 700
HOMEWORK QUIZ
 Divide the polynomials using synthetic division.
 (2x4 – x3 + 4) ÷ (x + 1)
 Factor the following polynomial. (x – 10) is one of
the factors.
 x3 – 12x2 + 12x + 80
Rational Zero Theorem
EXAMPLE 1
List possible rational zeros
List the possible rational zeros of f using the rational
zero theorem.
a.
f (x) = x3 + 2x2 – 11x + 12
Factors of the constant term: + 1, + 2, + 3, + 4, + 6, + 12
Factors of the leading coefficient: + 1
Possible rational zeros: +
,+
, +1
1
, +2 , + 3 , +
1
1
Simplified list of possible zeros: + 1, + 2, + 3, + 4, + 6, + 12
4
1
6
1
12
1
EXAMPLE 1
b.
List possible rational zeros
f (x) = 4x4 – x3 – 3x2 + 9x – 10
Factors of the constant term: + 1, + 2, + 5, + 10
Factors of the leading coefficient: + 1, + 2, + 4
Possible rational zeros:
+ 1, +
1
,2+
1
+ 5, +
2
10
,+
2
, + 5 , + 10 , +
1
1
, +1 , + 2
4
4
,1
2
2
2
5
4
+ 10
4
Simplified list of possible zeros: + 1, + 2, + 5, + 10, +
+, +5 , + 1
5
2
4
4
1
2
Finding all rational zeros
 Find all rational zeros for f(x) = x3 – 8x2 + 11x + 20.
 List all possible zeros. Then use synthetic division to
guess and check.
Find all rational zeros.
 Find all rational zeros for f(x) = x3 – 4x2 - 15x + 18.
Find all REAL zeros
 For more difficult examples, use a graphing
calculator to find reasonable estimates for rational
zeros.
 Find all real zeros for
f(x) = 10x4 – 11x3 – 42x2 + 7x + 12
Find all REAL zeros
 Find all real zeros for
f(x) = 2x4 + 5x3 – 18x2 – 19x + 42
EXAMPLE 4
Solve a multi-step problem
ICE SCULPTURES
Some ice sculptures are made
by filling a mold with water and
then freezing it. You are
making such an ice sculpture
for a school dance. It is to be
shaped like a pyramid with a
height that is 1 foot greater
than the length of each side of
its square base. The volume of
the ice sculpture is 4 cubic feet. What are the
dimensions of the mold?
Solve the following equation
 3x2 + 5x + 4 = 0
 How many solutions are there?
 What kind of solutions are they?
Fundamental Theorem of Algebra
 If f(x) is a polynomial of degree n, then the equation
f(x) = 0 has exactly n solutions, provided that
repeated solutions are counted multiple times.
 When is there a repeated solution?


When f(x) can be written with a factor raised to a power.
Ex: x2 – 8x + 16 = 0……..(x – 4)2 = 0
 What does the graph look like when there is a
repeated solution?

The graph touches the x-axis at that point, but does not cross
it.
GUIDED PRACTICE
for Example 2
Find all zeros of the polynomial function.
3.
f (x) = x3 + 7x2 + 15x + 9
ANSWER
The zeros of f are – 1, −3, and – 3.
4.
f (x) = x5 – 2x4 + 8x2 – 13x + 6
ANSWER
Zeros of f are 1, 1, – 2, 1 + i 2, and 1 – i 2
EXAMPLE 2
Find the zeros of a polynomial function
Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14.
SOLUTION
STEP 1
Find the rational zeros of f. Because f is a
polynomial function of degree 5, it has 5
zeros. The possible rational zeros are + 1, + 2,
+ 7, and + 14. Using synthetic division, you
can determine that – 1 is a zero repeated
twice and 2 is also a zero.
STEP 2
Write f (x) in factored form. Dividing f (x) by
its known factors x + 1, x + 1, and x – 2 gives a
quotient of x2 – 4x + 7. Therefore:
f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)
EXAMPLE 2
STEP 3
Find the zeros of a polynomial function
Find the complex zeros of f . Use the quadratic formula to
factor the trinomial into linear factors.
f(x) = (x + 1)2(x – 2) x – (2 + i 3 )
x – (2 – i
3 )
ANSWER
The zeros of f are – 1, – 1, 2, 2 + i 3 , and 2 – i 3.