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Math Problems
Precalculus
Pascal's triangle and the Binomial Theorem
Use Pascal's Triangle to complete the expansion of
(
p
+
q
)
4
(p+q)^4
(
p
+
q
)
4
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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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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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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
(
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
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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
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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
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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
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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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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
(
4
−
z
2
)
4
\left(4-z^{2}\right)^{4}
(
4
−
z
2
)
4
. Express your answer in simplest form.
\newline
Answer:
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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
(
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
(
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
−
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
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
+
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
(
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
(
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
(
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
(
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
(
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
+
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
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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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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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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If the projections of the legs onto the hypotenuse in a right triangle are, respectively,
6
c
m
6 \mathrm{~cm}
6
cm
and
12
c
m
12 \mathrm{~cm}
12
cm
, then the triangle's area is equal to:
\newline
(A)
18
c
m
2
18 \mathrm{~cm}^{2}
18
cm
2
\newline
(B)
18
2
c
m
2
18 \sqrt{ } 2 \mathrm{~cm}^{2}
18
2
cm
2
\newline
(C)
36
3
c
m
2
36 \sqrt{ } 3 \mathrm{~cm}^{2}
36
3
cm
2
\newline
(D)
18
3
c
m
2
18 \sqrt{3} \mathrm{~cm}^{2}
18
3
cm
2
\newline
(E)
54
2
c
m
2
54 \sqrt{ } 2 \mathrm{~cm}^{2}
54
2
cm
2
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