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Long
Division
Remainder
Theorem
Factor
Theorem
Synthetic
Division
Rational
Zeros
Theorem
Upper &
Lower
Bounds
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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
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