Long
Division
Remainder
Theorem
Factor
Theorem
Synthetic
Division
Rational
Zeros
Theorem
Upper &
Lower
Bounds
100
100
100
100
100
100
200
200
200
200
200
200
300
300
300
300
300
300
400
400
400
400
400
400
500
500
500
500
500
500
Division involves a dividend
divided by a divisor to obtain a
quotient and a _________.
remainder
4
3
2x -x -2
What is the answer of
divided by 2x2+x+1? Write in
polynomial form.
2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)
2
f(x)=x -2x+3
Divide
b x-1. Write
in polynomial form.
f(x)=(x=1)2+2
Divide f(x)=x3+4x2+7x-9 by x+3
and write a summary statement
in polynomial form.
f(x)=(x2+x+4)(x+3)-21
4
3
2
f(x)=x -2x +3x -4x+6
Divide
by
x2+2x-, write a summary
statement in fraction form.
f(x)=(x2-4x+12)(x2+2x-1)-32x+18
If a polynomial is divided by
___, then the remainder is
r=f(k).
x-k
Using the remainder theorem,
find the remainder when
f(x)=3x2+7x-20 is divided by
x+1.
r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24
What is the remainder when
f(x)=2x2-3x+1 is divided by x-2?
3
Use the remainder theorem to
find the remainder when you
divide f(x)=x3-x2+2x-1 by x+3
-43
Use the remainder theorem to
find the remainder when
f(x)=2x3-3x2+4x-7 by x-2
5
A polynomial function f(x) has a
factor x-k if and only if f(k)=_.
0
Using the factor theorem, factor
f(x)=3x2+7x-20 by dividing it by
the known factor x+4.
(3x-5)
Is x-1 a factor of x3-x2+x-1?
Yes
Use the factor theorem to
determine whether x-2 is a
3
factor of x +3x-4
No
Use the factor theorem to
determine where x+2 is a factor
3
2
of 4x +9x -3x-10
Yes
This shortcut method for the
division of a polynomial by a
______ _______ x-k is
synthetic division.
linear divisor
Divide 2x3-3x2-5x-12 by x-3
using synthetic division and
write a summary statement in
fraction form.
2x3-3x2-5x-12 = 2x2+3x+4
x-3
Divide using synthetic division
and write a summary statement
in fraction form:
x3-5x2+3x-2
x+1
2
X -6x+9+ -11
x+1
Divide using synthetic division
and write a summary statement
in fraction form:
9x3+7x2-3x
x-10
9x2+97x+967+ 9670
x-10
Divide using synthetic division
and write a summary statement
in fraction form:
5x4-3x+1
4-x
-5x3-20x2-80x-317+ -1269
4-x
Real zeros of a polynomial
function are either ________
zeros – zeroes that are rational
numbers – or __________
zeros – zeros that are irrational
numbers.
rational, irrational
Find the rational zeros of
f(x)=x3-3x2+1
f(1)=(1)3-3(1)2+1=-1
f(-1)=(-1)3-3(-1)2+1=-3
So f has no rational zeros.
Use the rational zeroes theorem
to write a list of all possible
3
rational zeroes of f(x)=6x -5x-1
1
1, 2, 3, 6
(all positive & negative)
Use the rational zeroes theorem
to write a list of possible rational
zeroes:
f(x)=2x3-x2-9x+9
1, 3, 9
1, 2
(all positive & negative)
Find all the zeroes of the
function and identify each zero
as rational or irrational:
f(x)=2x4-7x3-2x2-7x-4
-1/2 and 4; rational
No irrational
We narrow our search for real
zeros by using a test that
identifies upper and lower
______ for real zeros.
bounds
Find a lower bound of
f(x)=2x4-7x3-8x2+14x+8
-2
Use synthetic division to prove
that to number k is an upper
bound for the real zeroes of the
function:
k=3; f(x)=2x3-4x2+x+3
Yes
Use synthetic division to prove
that the number k is an upper
bound for the real zeroes of the
function f:
4
3
2
k=2; f(x)=x -x +x +x-12
Yes
Use synthetic division to prove
that the number k is a lower
bound for the real zeroes of the
function f:
2
2
k=-1; f(x)=3x -4x +x-2
Yes