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Math Problems
Algebra 2
Descartes' Rule of Signs
Let
a
a
a
,
b
b
b
,
c
c
c
be the three rational numbers where
a
=
2
3
a = \frac{2}{3}
a
=
3
2
,
b
=
4
5
b = \frac{4}{5}
b
=
5
4
and
c
=
−
5
6
c = -\frac{5}{6}
c
=
−
6
5
\newline
Verify:
a
+
(
b
+
c
)
=
(
a
+
b
)
+
c
a + (b + c) = (a + b) + c
a
+
(
b
+
c
)
=
(
a
+
b
)
+
c
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According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
f
(
x
)
=
x
4
+
5
x
3
+
2
x
2
−
4
x
−
5
f(x) = x^4 + 5x^3 + 2x^2 - 4x - 5
f
(
x
)
=
x
4
+
5
x
3
+
2
x
2
−
4
x
−
5
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
5
5
5
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
f
(
x
)
=
x
5
−
8
x
4
+
8
x
3
−
4
x
2
+
7
f(x) = x^5 - 8x^4 + 8x^3 - 4x^2 + 7
f
(
x
)
=
x
5
−
8
x
4
+
8
x
3
−
4
x
2
+
7
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
4
4
4
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
h
(
x
)
=
x
4
−
x
3
−
7
x
2
+
x
−
1
h(x) = x^4 - x^3 - 7x^2 + x - 1
h
(
x
)
=
x
4
−
x
3
−
7
x
2
+
x
−
1
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
2
2
2
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
h
(
x
)
=
x
6
−
6
x
3
−
4
x
2
−
6
x
+
1
h(x) = x^6 - 6x^3 - 4x^2 - 6x + 1
h
(
x
)
=
x
6
−
6
x
3
−
4
x
2
−
6
x
+
1
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
0
0
0
positive real zeros? Choose your answer based on the rule only.
\newline
h
(
x
)
=
x
3
−
3
x
2
+
8
x
+
6
h(x) = x^3 - 3x^2 + 8x + 6
h
(
x
)
=
x
3
−
3
x
2
+
8
x
+
6
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
3
+
6
x
2
−
x
−
9
g(x) = x^3 + 6x^2 - x - 9
g
(
x
)
=
x
3
+
6
x
2
−
x
−
9
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
4
4
4
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
h
(
x
)
=
x
4
+
5
x
3
+
5
x
2
+
4
x
+
1
h(x) = x^4 + 5x^3 + 5x^2 + 4x + 1
h
(
x
)
=
x
4
+
5
x
3
+
5
x
2
+
4
x
+
1
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
5
5
5
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
5
−
4
x
3
−
4
x
2
+
9
g(x) = x^5 - 4x^3 - 4x^2 + 9
g
(
x
)
=
x
5
−
4
x
3
−
4
x
2
+
9
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
6
+
6
x
3
+
8
x
2
+
x
−
6
g(x) = x^6 + 6x^3 + 8x^2 + x - 6
g
(
x
)
=
x
6
+
6
x
3
+
8
x
2
+
x
−
6
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
0
0
0
positive real zeros? Choose your answer based on the rule only.
\newline
f
(
x
)
=
x
5
+
2
x
4
+
8
x
3
−
4
x
2
+
5
f(x) = x^5 + 2x^4 + 8x^3 - 4x^2 + 5
f
(
x
)
=
x
5
+
2
x
4
+
8
x
3
−
4
x
2
+
5
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
2
2
2
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
4
−
x
3
−
8
x
2
−
6
x
+
8
g(x) = x^4 - x^3 - 8x^2 - 6x + 8
g
(
x
)
=
x
4
−
x
3
−
8
x
2
−
6
x
+
8
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
5
5
5
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
5
−
9
x
4
−
4
x
2
+
2
x
+
8
g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8
g
(
x
)
=
x
5
−
9
x
4
−
4
x
2
+
2
x
+
8
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
4
4
4
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
7
+
5
x
6
−
8
x
5
−
4
x
4
+
9
x
3
−
5
x
−
4
g(x) = x^7 + 5x^6 - 8x^5 - 4x^4 + 9x^3 - 5x - 4
g
(
x
)
=
x
7
+
5
x
6
−
8
x
5
−
4
x
4
+
9
x
3
−
5
x
−
4
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
6
+
2
x
5
+
6
x
4
+
x
−
8
g(x) = x^6 + 2x^5 + 6x^4 + x - 8
g
(
x
)
=
x
6
+
2
x
5
+
6
x
4
+
x
−
8
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
0
0
0
positive real zeros? Choose your answer based on the rule only.
\newline
h
(
x
)
=
x
6
+
2
x
5
+
3
x
4
+
9
x
3
+
7
x
2
+
7
h(x) = x^6 + 2x^5 + 3x^4 + 9x^3 + 7x^2 + 7
h
(
x
)
=
x
6
+
2
x
5
+
3
x
4
+
9
x
3
+
7
x
2
+
7
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
4
+
5
x
3
−
9
x
2
+
9
x
+
8
g(x) = x^4 + 5x^3 - 9x^2 + 9x + 8
g
(
x
)
=
x
4
+
5
x
3
−
9
x
2
+
9
x
+
8
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
5
5
5
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
5
+
5
x
4
+
5
x
3
+
2
x
2
+
6
g(x) = x^5 + 5x^4 + 5x^3 + 2x^2 + 6
g
(
x
)
=
x
5
+
5
x
4
+
5
x
3
+
2
x
2
+
6
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
g
(
x
)
=
x
7
−
5
x
6
−
7
x
5
−
3
x
2
+
2
x
−
2
g(x) = x^7 - 5x^6 - 7x^5 - 3x^2 + 2x - 2
g
(
x
)
=
x
7
−
5
x
6
−
7
x
5
−
3
x
2
+
2
x
−
2
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
1
1
1
positive real zero, including any repeated zeros? Choose your answer based on the rule only.
\newline
h
(
x
)
=
x
4
+
x
3
−
8
x
2
+
x
+
9
h(x) = x^4 + x^3 - 8x^2 + x + 9
h
(
x
)
=
x
4
+
x
3
−
8
x
2
+
x
+
9
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
4
4
4
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
f
(
x
)
=
x
4
+
x
3
−
3
x
2
+
5
x
+
5
f(x) = x^4 + x^3 - 3x^2 + 5x + 5
f
(
x
)
=
x
4
+
x
3
−
3
x
2
+
5
x
+
5
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
Evaluate
5
w
−
w
x
5 w-\frac{w}{x}
5
w
−
x
w
when
w
=
6
w=6
w
=
6
and
x
=
2
x=2
x
=
2
.
\newline
□
\square
□
Get tutor help
If
x
−
9
x-9
x
−
9
is a factor of the polynomial function
f
f
f
, what is the value of
f
(
9
)
f(9)
f
(
9
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
9
-9
−
9
\newline
(B)
0
0
0
\newline
(C)
9
9
9
\newline
(D) Cannot be determined from the given information
Get tutor help
If
x
+
7
x+7
x
+
7
is a factor of the polynomial function
g
g
g
, what is the value of
g
(
7
)
g(7)
g
(
7
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
7
-7
−
7
\newline
(B)
0
0
0
\newline
(C)
14
14
14
\newline
(D) Cannot be determined from the given information
Get tutor help
6
,
000
=
b
u
+
4
,
000
6,000=b u+4,000
6
,
000
=
b
u
+
4
,
000
\newline
Given the equation with constant
b
b
b
, what is the value of
600
,
000
+
100
b
u
+
400
,
000
600,000+100 b u+400,000
600
,
000
+
100
b
u
+
400
,
000
in millions?
\newline
(For example, if your answer is
12
12
12
.
4
4
4
million, enter
12
12
12
.
4
4
4
.)
Get tutor help
According to Descartes' Rule of Signs, can the polynomial function have exactly
5
5
5
positive real zeros, including any repeated zeros? Choose your answer based on the rule only.
\newline
f
(
x
)
=
x
7
+
8
x
5
+
x
4
+
x
3
+
8
x
2
−
7
x
−
5
f(x) = x^7 + 8x^5 + x^4 + x^3 + 8x^2 - 7x - 5
f
(
x
)
=
x
7
+
8
x
5
+
x
4
+
x
3
+
8
x
2
−
7
x
−
5
\newline
Choices:
\newline
[A]yes
\newline
[B]no
Get tutor help
