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Long Division
RemainderTheorem
Factor Theorem
SyntheticDivision
Rational Zeros
Theorem
Upper &Lower
Bounds
Division involves a dividend divided by a divisor to obtain a
quotient and a _________.
remainder
What is the answer of 2x4-x3-2 divided by 2x2+x+1? Write in
polynomial form.
2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)
Divide f(x)=x2-2x+3 b x-1. Write in polynomial form.
f(x)=(x=1)2+2
f(x)=(x2+x+4)(x+3)-21
Divide f(x)=x3+4x2+7x-9 by x+3 and write a summary statement
in polynomial form.
Divide f(x)=x4-2x3+3x2-4x+6 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
5
Use the remainder theorem to find the remainder when f(x)=2x3-3x2+4x-7 by x-2
0
A polynomial function f(x) has a factor x-k if and only if f(k)=_.
(3x-5)
Using the factor theorem, factor f(x)=3x2+7x-20 by dividing it by
the known factor x+4.
Yes
Is x-1 a factor of x3-x2+x-1?
No
Use the factor theorem to determine whether x-2 is a
factor of x3+3x-4
Yes
Use the factor theorem to determine where x+2 is a factor
of 4x3+9x2-3x-10
linear divisor
This shortcut method for the division of a polynomial by a
______ _______ x-k is synthetic division.
2x3-3x2-5x-12 = 2x2+3x+4
x-3
Divide 2x3-3x2-5x-12 by x-3 using synthetic division and
write a summary statement in fraction form.
X2-6x+9+ -11
x+1
Divide using synthetic division and write a summary statement
in fraction form: x3-5x2+3x-2
x+1
9x2+97x+967+ 9670
x-10
Divide using synthetic division and write a summary statement
in fraction form: 9x3+7x2-3x
x-10
-5x3-20x2-80x-317+ -1269
4-x
Divide using synthetic division and write a summary statement
in fraction form: 5x4-3x+1
4-x
rational, irrational
Real zeros of a polynomial function are either ________
zeros – zeroes that are rational numbers – or __________
zeros – zeros that are irrational numbers.
f(1)=(1)3-3(1)2+1=-1
f(-1)=(-1)3-3(-1)2+1=-3
So f has no rational zeros.
Find the rational zeros of f(x)=x3-3x2+1
1
1, 2, 3, 6
(all positive & negative)
Use the rational zeroes theorem to write a list of all possible
rational zeroes of f(x)=6x3-5x-1
1, 3, 9
1, 2
(all positive & negative)
Use the rational zeroes theorem to write a list of possible rational
zeroes:f(x)=2x3-x2-9x+9
-1/2 and 4; rational
No irrational
Find all the zeroes of the function and identify each zero
as rational or irrational:f(x)=2x4-7x3-2x2-7x-4
bounds
We narrow our search for real zeros by using a test that identifies upper and lower
______ for real zeros.
-2
Find a lower bound of f(x)=2x4-7x3-8x2+14x+8
Yes
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:
k=2; f(x)=x4-x3+x2+x-12
Yes
Use synthetic division to prove that the number k is a lower
bound for the real zeroes of the function f:
k=-1; f(x)=3x2-4x2+x-2
