Use one of the appropriate patterns $( a + b ) ^ { 2 } = a ^ { 2 } + 2 a b + b ^ { 2 }$ ; $( a - b ) ^ { 2 } = a ^ { 2 } - 2 a b + b ^ { 2 }$ ; $( a + b ) ( a - b ) = a ^ { 2 } - b ^ { 2 }$ to find the following product. $- 9 x ( 4 - 9 x ) ( 4 + 9 x )$
A) $- 36 x + 729 x ^ { 3 }$
B) $- 144 x + 729 x ^ { 3 }$
C) $- 144 x - 729 x ^ { 2 }$
D) $- 144 x + 81 x ^ { 3 }$
E) none of these

Question

Quizplus · Accepted Answer

First, we can simplify the expression by using the pattern $(a+b)(a-b) = a^2-b^2)$ to get:

\begin{align*}
-9x(4-9x)(4+9x) &= -9x[(4)^2-(9x)^2]\
&=-9x[16-81x^2]\
&=-9x(4^2-(9x)^2)\
&=-9x(4-9x)(4+9x)
\end{align*}

Now, we can use the pattern $(a-b)^2 = a^2 - 2ab + b^2$ to expand $(4-9x)^2$ and $(4+9x)^2$:

\begin{align*}
(4-9x)^2 &= 4^2 - 2(4)(9x) + (9x)^2\
&= 16 - 72x + 81x^2
\end{align*}

\begin{align*}
(4+9x)^2 &= 4^2 + 2(4)(9x) + (9x)^2\
&= 16 + 72x + 81x^2
\end{align*}

Plugging these expressions back into our original equation, we have:

\begin{align*}
-9x(16-72x+81x^2)(16+72x+81x^2) &= -9x[(16^2-(72x)^2+81x^2)][(16^2+(72x)^2+81x^2)]\
&= -9x[(256-5184x^2+6561x^4)][(256+5184x^2+6561x^4)]\
&= -9x(256^2 - (5184x^2)^2 + (6561x^4)^2)\
&= -9x(256^2 - 5184^2 x^4 + 6561^2 x^8)\
&= -9x(1679616 - 2985984 x^4 + 43046721 x^8)
\end{align*}

Thus, the final answer is $-144x+729x^3$, which corresponds to choice B.

Use One of the Appropriate Patterns (a+b)2=a2+2ab+b2( a + b ) ^ { 2 } = a ^ { 2 } + 2 a b + b ^ { 2 }(a+b)2=a2+2ab+b2

Use One of the Appropriate Patterns $( a + b ) ^ { 2 } = a ^ { 2 } + 2 a b + b ^ { 2 }$