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Proving Polynomial Identities
Choose Course
Algebra 2
Polynomial Functions
Proving Polynomial Identities
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Proving Polynomial Identities 1.6 - Solution
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a
The
difference of squares
has the form
(
a
−
b
)
(
a
+
b
)
=
a
2
−
b
2
(a - b)(a + b) = a^2 - b^2
(
a
−
b
)
(
a
+
b
)
=
a
2
−
b
2
and we see that our expression has the same form, where
a
=
p
a=p
a
=
p
and
b
=
64
b = 64
b
=
6
4
.
(
p
−
64
)
(
p
+
64
)
\left(p-64\right)\left(p+64\right)
(
p
−
6
4
)
(
p
+
6
4
)
ExpandDiffSquares
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
(a+b)(a-b)=a^2-b^2
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
p
2
−
6
4
2
p^2 - 64^2
p
2
−
6
4
2
CalcPow
Calculate power
p
2
−
4096
p^2 - 4096
p
2
−
4
0
9
6
b
The
difference of squares
has the form
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
(a + b)(a - b) = a^2 - b^2
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
and we see that our expression has the same form, with
a
=
x
a=x
a
=
x
and
b
=
2
y
b = 2y
b
=
2
y
.
(
x
+
2
y
)
(
x
−
2
y
)
(x+2y)(x-2y)
(
x
+
2
y
)
(
x
−
2
y
)
ExpandDiffSquares
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
(a+b)(a-b)=a^2-b^2
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
x
2
−
(
2
y
)
2
x^2 - (2y)^2
x
2
−
(
2
y
)
2
PowProdII
(
a
b
)
m
=
a
m
b
m
\left(a b\right)^{m}=a^m b^m
(
a
b
)
m
=
a
m
b
m
x
2
−
2
2
y
2
x^2 - 2^2y^2
x
2
−
2
2
y
2
CalcPow
Calculate power
x
2
−
4
y
2
x^2 - 4y^2
x
2
−
4
y
2
c
We proceed in the same way to rewrite the expression with the difference of two squares. Note that the square of
x
2
x^2
x
2
is
(
x
2
)
2
.
(x^2)^2.
(
x
2
)
2
.
(
x
2
−
8
)
(
x
2
+
8
)
\left(x^2-8\right)\left(x^2+8\right)
(
x
2
−
8
)
(
x
2
+
8
)
ExpandDiffSquares
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
(a+b)(a-b)=a^2-b^2
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
(
x
2
)
2
−
8
2
\left(x^2\right)^2 - 8^2
(
x
2
)
2
−
8
2
PowPow
(
a
m
)
n
=
a
m
⋅
n
\left(a^{m}\right)^{n}=a^{m\cdot n}
(
a
m
)
n
=
a
m
⋅
n
x
4
−
8
2
x^4 - 8^2
x
4
−
8
2
CalcPow
Calculate power
x
4
−
64
x^4 - 64
x
4
−
6
4
