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Math Problems
Precalculus
Pascal's triangle and the Binomial Theorem
The binomial expansion of
(
x
2
+
y
)
2
\left(x^{2}+y\right)^{2}
(
x
2
+
y
)
2
is
\qquad
\newline
A.
x
2
+
2
x
2
y
+
y
2
x^{2}+2 x^{2} y+y^{2}
x
2
+
2
x
2
y
+
y
2
\newline
B.
x
4
+
2
x
3
y
+
x
2
y
2
x^{4}+2 x^{3} y+x^{2} y^{2}
x
4
+
2
x
3
y
+
x
2
y
2
\newline
C.
x
4
+
2
x
2
y
+
y
2
x^{4}+2 x^{2} y+y^{2}
x
4
+
2
x
2
y
+
y
2
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Approximating Irrational Numbers - Tutorial - Part
1
1
1
- Level H
\newline
Let's try to find a rational approximation for
2
\sqrt{2}
2
.
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Find the geometric location of the points of the process
x
2
+
2
y
2
+
3
z
2
+
x
y
+
2
x
z
+
4
y
z
=
8
x^{2}+2y^{2}+3z^{2}+xy+2xz+4yz=8
x
2
+
2
y
2
+
3
z
2
+
x
y
+
2
x
z
+
4
yz
=
8
where the tangent plane is parallel to the
x
y
xy
x
y
plane
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Search res
\newline
Facebook
\newline
dodge car
\newline
DieHard P
\newline
car batter
\newline
(
1568
1568
1568
) Tain
\newline
Copy of
8
8
8
\newline
Scienti:
\newline
FLVS
\newline
B Lessons
\newline
Assessments
\newline
Gradebook
\newline
Email
1
1
1
\newline
Tools
\newline
The quadratic functions
f
(
x
)
\mathrm{f}(\mathrm{x})
f
(
x
)
and
g
(
x
)
\mathrm{g}(\mathrm{x})
g
(
x
)
are described in the table.
\newline
\begin{tabular}{|c|c|c|}
\newline
\hline
x
x
x
&
f
(
x
)
f(x)
f
(
x
)
&
g
(
x
)
g(x)
g
(
x
)
\\
\newline
\hline
−
6
-6
−
6
&
36
36
36
&
4
4
4
\\
\newline
\hline
−
5
-5
−
5
&
25
25
25
&
1
1
1
\\
\newline
\hline
−
4
-4
−
4
&
16
16
16
&
0
0
0
\\
\newline
\hline
−
3
-3
−
3
&
9
9
9
&
1
1
1
\\
\newline
\hline
−
2
-2
−
2
&
4
4
4
&
4
4
4
\\
\newline
\hline
−
1
-1
−
1
&
1
1
1
&
9
9
9
\\
\newline
\hline
0
0
0
&
0
0
0
&
16
16
16
\\
\newline
\hline
1
1
1
&
1
1
1
&
25
25
25
\\
\newline
\hline
2
2
2
&
4
4
4
&
36
36
36
\\
\newline
\hline
\newline
\end{tabular}
\newline
In which direction and by how many units should
f
(
x
)
f(x)
f
(
x
)
be shifted to match
g
(
x
)
g(x)
g
(
x
)
?
\newline
Left by
4
4
4
units
\newline
Right by
4
4
4
units
\newline
Left by
8
8
8
units
\newline
Right by
8
8
8
units
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Use Pascal's Triangle to complete the expansion of
(
u
+
v
)
4
(u + v)^4
(
u
+
v
)
4
.
\newline
u
4
+
4
u
3
v
+
‾
u
2
v
2
+
4
u
v
3
+
v
4
u^4 + 4u^3v + \underline{\hspace{2em}}u^2v^2 + 4uv^3 + v^4
u
4
+
4
u
3
v
+
u
2
v
2
+
4
u
v
3
+
v
4
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Use Pascal's Triangle to complete the expansion of
(
u
+
v
)
3
(u + v)^3
(
u
+
v
)
3
.
\newline
u
3
+
3
u
2
v
+
‾
u
v
2
+
v
3
u^3 + 3u^2v + \underline{\hspace{1cm}}uv^2 + v^3
u
3
+
3
u
2
v
+
u
v
2
+
v
3
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Use Pascal's Triangle to complete the expansion of
(
p
+
q
)
5
(p + q)^5
(
p
+
q
)
5
.
\newline
p
5
+
p^5 +
p
5
+
____
p
4
q
+
10
p
3
q
2
+
10
p
2
q
3
+
5
p
q
4
+
q
5
p^4q + 10p^3q^2 + 10p^2q^3 + 5pq^4 + q^5
p
4
q
+
10
p
3
q
2
+
10
p
2
q
3
+
5
p
q
4
+
q
5
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Use Pascal's Triangle to complete the expansion of
(
s
+
t
)
3
(s + t)^3
(
s
+
t
)
3
.
\newline
s
3
+
s^3 +
s
3
+
____
s
2
t
+
3
s
t
2
+
t
3
s^2t + 3st^2 + t^3
s
2
t
+
3
s
t
2
+
t
3
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Use Pascal's Triangle to complete the expansion of
(
w
+
x
)
6
(w + x)^6
(
w
+
x
)
6
.
\newline
w
6
+
6
w
5
x
+
_
_
_
_
w
4
x
2
+
20
w
3
x
3
+
15
w
2
x
4
+
6
w
x
5
+
x
6
w^6 + 6w^5x + \_\_\_\_w^4x^2 + 20w^3x^3 + 15w^2x^4 + 6wx^5 + x^6
w
6
+
6
w
5
x
+
____
w
4
x
2
+
20
w
3
x
3
+
15
w
2
x
4
+
6
w
x
5
+
x
6
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Use Pascal's Triangle to complete the expansion of
\newline
(
p
+
q
)
6
(p + q)^6
(
p
+
q
)
6
.
p
6
+
p^6 +
p
6
+
____
p
5
q
+
15
p
4
q
2
+
20
p
3
q
3
+
15
p
2
q
4
+
6
p
q
5
+
q
6
p^5q + 15p^4q^2 + 20p^3q^3 + 15p^2q^4 + 6pq^5 + q^6
p
5
q
+
15
p
4
q
2
+
20
p
3
q
3
+
15
p
2
q
4
+
6
p
q
5
+
q
6
Get tutor help
Use Pascal's Triangle to complete the expansion of
(
v
+
w
)
3
(v + w)^3
(
v
+
w
)
3
.
\newline
v
3
+
v^3 +
v
3
+
____
v
2
w
+
3
v
w
2
+
w
3
v^2w + 3vw^2 + w^3
v
2
w
+
3
v
w
2
+
w
3
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Use Pascal's Triangle to complete the expansion of
(
x
+
y
)
5
(x + y)^5
(
x
+
y
)
5
.
\newline
x
5
+
5
x
4
y
+
10
x
3
y
2
+
‾
x
2
y
3
+
5
x
y
4
+
y
5
x^5 + 5x^4y + 10x^3y^2 + \underline{\quad}x^2y^3 + 5xy^4 + y^5
x
5
+
5
x
4
y
+
10
x
3
y
2
+
x
2
y
3
+
5
x
y
4
+
y
5
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Use Pascal's Triangle to complete the expansion of
(
t
+
u
)
4
(t + u)^4
(
t
+
u
)
4
.
\newline
t
4
+
t^4 +
t
4
+
____
t
3
u
+
6
t
2
u
2
+
4
t
u
3
+
u
4
t^3u + 6t^2u^2 + 4tu^3 + u^4
t
3
u
+
6
t
2
u
2
+
4
t
u
3
+
u
4
Get tutor help
Use Pascal's Triangle to complete the expansion of
(
s
+
t
)
3
(s + t)^3
(
s
+
t
)
3
.
\newline
s
3
+
3
s
2
t
+
‾
s
t
2
+
t
3
s^3 + 3s^2t + \underline{\hspace{3em}}st^2 + t^3
s
3
+
3
s
2
t
+
s
t
2
+
t
3
Get tutor help
Use Pascal's Triangle to complete the expansion of
(
v
+
w
)
4
(v + w)^4
(
v
+
w
)
4
.
\newline
v
4
+
v^4 +
v
4
+
____
v
3
w
+
6
v
2
w
2
+
4
v
w
3
+
w
4
v^3w + 6v^2w^2 + 4vw^3 + w^4
v
3
w
+
6
v
2
w
2
+
4
v
w
3
+
w
4
Get tutor help
Use Pascal's Triangle to complete the expansion of
(
r
+
s
)
4
(r + s)^4
(
r
+
s
)
4
.
\newline
r
4
+
4
r
3
s
+
‾
r
2
s
2
+
4
r
s
3
+
s
4
r^4 + 4r^3s + \underline{\hspace{2em}}r^2s^2 + 4rs^3 + s^4
r
4
+
4
r
3
s
+
r
2
s
2
+
4
r
s
3
+
s
4
Get tutor help
Use Pascal's Triangle to complete the expansion of
(
y
+
z
)
4
(y + z)^4
(
y
+
z
)
4
.
\newline
y
4
+
y^4 +
y
4
+
____
y
3
z
+
6
y
2
z
2
+
4
y
z
3
+
z
4
y^3z + 6y^2z^2 + 4yz^3 + z^4
y
3
z
+
6
y
2
z
2
+
4
y
z
3
+
z
4
Get tutor help
20
20
20
. Find the L.C.M. of following algebraic terms,
\newline
2
x
,
3
x
2
,
x
y
2 x, 3 x^{2}, x y
2
x
,
3
x
2
,
x
y
Get tutor help
the fourth term in the expansion of
(
a
x
+
2
)
10
(ax + \sqrt{2}) ^ {10}
(
a
x
+
2
)
10
is
30
x
7
10
30x ^ 7 10
30
x
7
10
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Can's
\newline
Playing
\newline
ClassLink
\newline
Download..
\newline
d...
\newline
Algebra C\&C Milestone Practice Section
1
1
1
\newline
Question
1
1
1
\newline
Pause
\newline
Review
\newline
Which function can be used to model the data in this table?
\newline
\begin{tabular}{|c|c|}
\newline
\hline
x
x
x
&
f
(
x
)
f(x)
f
(
x
)
\\
\newline
\hline
0
0
0
&
−
1
-1
−
1
\\
\newline
\hline
2
2
2
&
0
0
0
\\
\newline
\hline
6
6
6
&
2
2
2
\\
\newline
\hline
\newline
\end{tabular}
\newline
ABG
\newline
A.
f
(
x
)
=
3
x
f(x)=3 x
f
(
x
)
=
3
x
\newline
B.
f
(
x
)
=
x
2
−
1
f(x)=\frac{x}{2}-1
f
(
x
)
=
2
x
−
1
\newline
C.
f
(
x
)
=
x
−
2
f(x)=x-2
f
(
x
)
=
x
−
2
\newline
D.
f
(
x
)
=
2
x
−
1
f(x)=2 x-1
f
(
x
)
=
2
x
−
1
Get tutor help
Use Pascal's Triangle to expand
(
4
y
2
+
1
)
3
\left(4 y^{2}+1\right)^{3}
(
4
y
2
+
1
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
4
−
z
2
)
4
\left(4-z^{2}\right)^{4}
(
4
−
z
2
)
4
. Express your answer in simplest form.
\newline
Answer:
Get tutor help
Use Pascal's Triangle to expand
(
x
2
−
z
2
)
4
\left(x^{2}-z^{2}\right)^{4}
(
x
2
−
z
2
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
z
2
+
5
x
2
)
4
\left(z^{2}+5 x^{2}\right)^{4}
(
z
2
+
5
x
2
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
2
−
5
y
)
4
(2-5 y)^{4}
(
2
−
5
y
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
−
3
y
)
5
(3 z-3 y)^{5}
(
3
z
−
3
y
)
5
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
x
2
+
2
z
)
4
\left(x^{2}+2 z\right)^{4}
(
x
2
+
2
z
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
y
−
4
z
)
4
(y-4 z)^{4}
(
y
−
4
z
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
4
+
2
y
)
3
(4+2 y)^{3}
(
4
+
2
y
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
−
2
y
)
5
(3 z-2 y)^{5}
(
3
z
−
2
y
)
5
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
5
y
2
+
x
2
)
3
\left(5 y^{2}+x^{2}\right)^{3}
(
5
y
2
+
x
2
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
2
y
+
x
)
5
(2 y+x)^{5}
(
2
y
+
x
)
5
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
4
y
2
+
5
x
2
)
4
\left(4 y^{2}+5 x^{2}\right)^{4}
(
4
y
2
+
5
x
2
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
+
y
2
)
5
\left(3+y^{2}\right)^{5}
(
3
+
y
2
)
5
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
2
−
4
x
)
3
\left(3 z^{2}-4 x\right)^{3}
(
3
z
2
−
4
x
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
−
5
x
2
)
4
\left(3-5 x^{2}\right)^{4}
(
3
−
5
x
2
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
x
−
5
z
)
3
(3 x-5 z)^{3}
(
3
x
−
5
z
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
y
+
5
z
)
4
(3 y+5 z)^{4}
(
3
y
+
5
z
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
y
2
−
2
)
3
\left(3 y^{2}-2\right)^{3}
(
3
y
2
−
2
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
2
+
5
)
4
\left(3 z^{2}+5\right)^{4}
(
3
z
2
+
5
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
x
+
z
)
4
(x+z)^{4}
(
x
+
z
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
−
4
)
4
(3 z-4)^{4}
(
3
z
−
4
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
2
+
5
x
2
)
4
\left(3 z^{2}+5 x^{2}\right)^{4}
(
3
z
2
+
5
x
2
)
4
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
2
y
2
+
z
2
)
3
\left(2 y^{2}+z^{2}\right)^{3}
(
2
y
2
+
z
2
)
3
. Express your answer in simplest form.
\newline
Answer:
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Use Pascal's Triangle to expand
(
3
z
2
−
4
)
3
\left(3 z^{2}-4\right)^{3}
(
3
z
2
−
4
)
3
. Express your answer in simplest form.
\newline
Answer:
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Question
7
7
7
/
91
91
91
\newline
(
1
1
1
)
\newline
NEXT
\newline
BOOKMARK
\newline
CHECK ANSWE
\newline
7
7
7
In which expression is the coefficient of the
n
\mathrm{n}
n
term
−
1
-1
−
1
?
\newline
(A)
3
n
2
−
4
n
−
1
3 n^{2}-4 n-1
3
n
2
−
4
n
−
1
\newline
(B)
−
n
2
+
5
n
+
4
-n^{2}+5 n+4
−
n
2
+
5
n
+
4
\newline
(C)
−
2
n
2
−
n
+
5
-2 n^{2}-n+5
−
2
n
2
−
n
+
5
\newline
(D)
4
n
2
+
n
−
5
4 n^{2}+n-5
4
n
2
+
n
−
5
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A triangle has side lengths of
(
1.3
w
−
9.9
)
(1.3 w-9.9)
(
1.3
w
−
9.9
)
centimeters,
(
3.6
w
+
7.2
)
(3.6 w+7.2)
(
3.6
w
+
7.2
)
centimeters, and
(
5.9
x
−
2.8
)
(5.9 x-2.8)
(
5.9
x
−
2.8
)
centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
\newline
−
8.6
w
+
3.1
x
+
10.8
-8.6 w+3.1 x+10.8
−
8.6
w
+
3.1
x
+
10.8
\newline
−
5.5
+
4.9
w
+
5.9
x
-5.5+4.9 w+5.9 x
−
5.5
+
4.9
w
+
5.9
x
\newline
3.1
x
+
2.2
w
3.1 x+2.2 w
3.1
x
+
2.2
w
\newline
−
2.7
+
4.9
w
+
3.1
x
-2.7+4.9 w+3.1 x
−
2.7
+
4.9
w
+
3.1
x
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1
1
1
. (
3
3
3
pts) For the function
f
(
x
)
=
{
2
x
if
x
<
1
−
4
x
if
x
≥
1
f(x)=\left\{\begin{array}{cl}2 x & \text { if } x<1 \\ -4 x & \text { if } x \geq 1\end{array}\right.
f
(
x
)
=
{
2
x
−
4
x
if
x
<
1
if
x
≥
1
, evaluate the left and right limits using the table shown below:
\newline
\begin{tabular}{cccl}
\newline
x
x
x
&
2
x
2 x
2
x
&
x
x
x
&
−
4
x
-4 x
−
4
x
\\
\newline
\hline
0
0
0
.
9
9
9
&
1
1
1
.
8
8
8
&
1
1
1
.
1
1
1
&
−
4
-4
−
4
.
4
4
4
\\
\newline
\hline
0
0
0
.
99
99
99
&
1
1
1
.
98
98
98
&
1
1
1
.
01
01
01
&
−
4
-4
−
4
.
04
04
04
\\
\newline
\hline
0
0
0
.
999
999
999
&
1
1
1
.
998
998
998
&
1
1
1
.
001
001
001
&
−
4
-4
−
4
.
004
004
004
\\
\newline
\hline
0
0
0
.
9999
9999
9999
&
1
1
1
.
9998
9998
9998
&
1
1
1
.
0001
0001
0001
&
−
4
-4
−
4
.
0004
0004
0004
\\
\newline
\hline
0
0
0
.
99999
99999
99999
&
1
1
1
.
99998
99998
99998
&
1
1
1
.
00001
00001
00001
&
−
4
-4
−
4
.
00004
00004
00004
\\
\newline
\hline
\newline
\end{tabular}
\newline
a)
lim
x
→
1
−
f
(
x
)
\lim _{x \rightarrow 1^{-}} f(x)
lim
x
→
1
−
f
(
x
)
\newline
b)
lim
x
→
1
+
f
(
x
)
\lim _{x \rightarrow 1^{+}} f(x)
lim
x
→
1
+
f
(
x
)
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Determine the exact intersection point of lines
3
y
=
6
x
−
5
3y=6x-5
3
y
=
6
x
−
5
and
3
x
=
7
y
−
109
3
3x=7y-\frac{109}{3}
3
x
=
7
y
−
3
109
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Find the coefficient of the
x
2
x^{2}
x
2
term in the binomial expansion of
(
4
x
+
2
)
4
(4 x+2)^{4}
(
4
x
+
2
)
4
.
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1
2
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