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Precalculus
Pascal's triangle and the Binomial Theorem
Use Pascal's Triangle to complete the expansion of
(
y
−
z
)
7
(y-z)^{7}
(
y
−
z
)
7
.
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Use Pascal's Triangle to complete the expansion of
(
t
−
u
)
5
(t-u)^{5}
(
t
−
u
)
5
.
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Use Pascal's Triangle to complete the expansion of
(
t
−
u
)
5
(t-u)^{5}
(
t
−
u
)
5
.
\newline
□
\square
□
+
+
+
□
\square
□
+
10
t
3
u
2
−
10
t
2
u
3
+
5
t
u
4
−
u
5
+10 t^{3} u^{2}-10 t^{2} u^{3}+5 t u^{4}-u^{5}
+
10
t
3
u
2
−
10
t
2
u
3
+
5
t
u
4
−
u
5
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Use Pascal's Triangle to expand
(
v
+
w
)
3
(v+w)^3
(
v
+
w
)
3
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Use Pascal's Triangle to expand
(
q
+
r
)
2
(q+r)^2
(
q
+
r
)
2
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Use Pascal's Triangle to complete the expansion of
(
u
−
v
)
3
(u-v)^3
(
u
−
v
)
3
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Use Pascal's Triangle to complete the expansion of
(
t
+
u
)
3
(t+u)^3
(
t
+
u
)
3
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Use Pascal's Triangle to complete the expansion of
(
q
–
r
)
6
(q–r)^6
(
q
–
r
)
6
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Use Pascal's Triangle to complete the expansion of
(
r
–
s
)
4
(r–s)^4
(
r
–
s
)
4
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Use Pascal's Triangle to complete the expansion of
(
x
+
y
)
6
(x+y)^6
(
x
+
y
)
6
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Use Pascal's Triangle to complete the expansion of
(
r
+
s
)
4
(r+s)^4
(
r
+
s
)
4
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Use Pascal's Triangle to complete the expansion of
(
q
–
r
)
7
(q–r)^7
(
q
–
r
)
7
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Use Pascal's Triangle to complete the expansion of
(
p
–
q
)
6
(p–q)^6
(
p
–
q
)
6
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Use Pascal's Triangle to complete the expansion of
(
q
−
r
)
3
(q-r)^3
(
q
−
r
)
3
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Use Pascal's Triangle to complete the expansion of
(
q
–
r
)
5
(q–r)^5
(
q
–
r
)
5
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Use Pascal's Triangle to complete the expansion of
(
q
–
r
)
4
(q–r)^4
(
q
–
r
)
4
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Use Pascal's Triangle to complete the expansion of
(
p
+
q
)
4
(p+q)^4
(
p
+
q
)
4
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Use Pascal's Triangle to complete the expansion of
(
p
+
q
)
4
(p+q)^4
(
p
+
q
)
4
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Use Pascal's Triangle to complete the expansion of
(
u
+
v
)
6
(u+v)^6
(
u
+
v
)
6
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Use Pascal's Triangle to complete the expansion of
(
y
+
z
)
3
(y+z)^3
(
y
+
z
)
3
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Use Pascal's Triangle to complete the expansion of
(
w
+
x
)
5
(w+x)^5
(
w
+
x
)
5
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Use Pascal's Triangle to complete the expansion of
(
t
+
u
)
7
(t+u)^7
(
t
+
u
)
7
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Use Pascal's Triangle to complete the expansion of
(
r
+
s
)
2
(r+s)^2
(
r
+
s
)
2
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Use Pascal's Triangle to complete the expansion of
(
y
+
z
)
4
(y+z)^4
(
y
+
z
)
4
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Use Pascal's Triangle to complete the expansion of
(
y
+
z
)
4
(y+z)^4
(
y
+
z
)
4
.
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Use Pascal's Triangle to complete the expansion of
(
y
+
z
)
4
(y+z)^4
(
y
+
z
)
4
.
y
4
+
4
y
3
z
+
6
y
2
z
2
+
y
z
3
+
z
4
y^4+4y^3z+6y^2z^2+ yz^3+z^4
y
4
+
4
y
3
z
+
6
y
2
z
2
+
y
z
3
+
z
4
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docs.google.com/forms/d/e/
1
1
1
FAlpQLScMiHtBw
1
1
1
q
4
4
4
DatHWwyDPKDUsXG-BQ
7
7
7
BNinWlynLLPgGC
67
67
67
nOw/formResponse
\newline
7
7
7
. *
\newline
2
2
2
points
\newline
7
7
7
. What is the least common denominator of the expressions below?
\newline
−
2
p
+
1
p
2
−
5
p
+
6
,
p
+
3
2
−
p
-\frac{2 p+1}{p^{2}-5 p+6}, \quad \frac{p+3}{2-p}
−
p
2
−
5
p
+
6
2
p
+
1
,
2
−
p
p
+
3
\newline
A.
p
−
2
p-2
p
−
2
\newline
C.
p
2
−
5
p
+
6
p^{2}-5 p+6
p
2
−
5
p
+
6
\newline
B.
2
p
−
5
2 p-5
2
p
−
5
\newline
D.
(
2
−
p
)
(
p
2
−
5
p
+
6
)
(2-p)\left(p^{2}-5 p+6\right)
(
2
−
p
)
(
p
2
−
5
p
+
6
)
\newline
Choose
\newline
8
8
8
. *
\newline
2
2
2
points
\newline
8
8
8
. What is the least common denominator of the expressions below?
\newline
3
x
−
4
2
,
x
−
1
2
,
2
2
\frac{3 x-4}{2}, \quad \frac{x-1}{2}, \quad \frac{2}{2}
2
3
x
−
4
,
2
x
−
1
,
2
2
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What is the
4
4
4
th term of the expansion of
(
1
−
2
x
)
n
(1-2 x)^{n}
(
1
−
2
x
)
n
if the binomial coefficients are taken from the row of Pascal's triangle shown below?
\newline
1
6
15
20
15
6
1
\begin{array}{lllllll} 1 & 6 & 15 & 20 & 15 & 6 & 1 \end{array}
1
6
15
20
15
6
1
\newline
240
x
4
240 x^{4}
240
x
4
\newline
−
20
x
3
-20 x^{3}
−
20
x
3
\newline
−
160
x
3
-160 x^{3}
−
160
x
3
\newline
160
x
3
160 x^{3}
160
x
3
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Teach me how to do this problem in english: Calcula el \'area de la superficie esf\'erica:
x
2
+
y
2
+
z
2
=
16
x^2 + y^2 + z^2 = 16
x
2
+
y
2
+
z
2
=
16
comprendida entre los planos:
z
=
2
z = 2
z
=
2
,
z
=
1
z = 1
z
=
1
.
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16
16
16
Read the definition. Then choose the vocabulary term defined.
\newline
The broad, flat region that stretches between the Mississippi River and the Rocky Mountains in North America
\newline
A Great Plains
\newline
B Great Plateau
\newline
C Great Basil
\newline
D Death Valley
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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
(
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
Get tutor help
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
Get tutor help
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
Get tutor help
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
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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
(
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
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
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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
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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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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
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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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Algebra C\&C Milestone Practice Section
1
1
1
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Question
1
1
1
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Pause
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Review
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Which function can be used to model the data in this table?
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\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
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\end{tabular}
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ABG
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A.
f
(
x
)
=
3
x
f(x)=3 x
f
(
x
)
=
3
x
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B.
f
(
x
)
=
x
2
−
1
f(x)=\frac{x}{2}-1
f
(
x
)
=
2
x
−
1
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C.
f
(
x
)
=
x
−
2
f(x)=x-2
f
(
x
)
=
x
−
2
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D.
f
(
x
)
=
2
x
−
1
f(x)=2 x-1
f
(
x
)
=
2
x
−
1
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