Question
Match the following polynomial identity to its expanded form.
x^4-a^4
x^4-a^4
Answers
GPT-4o mini
The polynomial identity \( x^4 - a^4 \) can be factored using the difference of squares. It can be expressed as:
\[
x^4 - a^4 = (x^2 - a^2)(x^2 + a^2)
\]
Next, we can factor \( x^2 - a^2 \) further since it is also a difference of squares:
\[
x^2 - a^2 = (x - a)(x + a)
\]
Putting it all together, we have:
\[
x^4 - a^4 = (x - a)(x + a)(x^2 + a^2)
\]
So, the expanded form of \( x^4 - a^4 \) is:
\[
(x - a)(x + a)(x^2 + a^2)
\]
Thus, the matching factored form of the polynomial identity \( x^4 - a^4 \) to its expanded form is:
\[
x^4 - a^4 = (x - a)(x + a)(x^2 + a^2)
\]
\[
x^4 - a^4 = (x^2 - a^2)(x^2 + a^2)
\]
Next, we can factor \( x^2 - a^2 \) further since it is also a difference of squares:
\[
x^2 - a^2 = (x - a)(x + a)
\]
Putting it all together, we have:
\[
x^4 - a^4 = (x - a)(x + a)(x^2 + a^2)
\]
So, the expanded form of \( x^4 - a^4 \) is:
\[
(x - a)(x + a)(x^2 + a^2)
\]
Thus, the matching factored form of the polynomial identity \( x^4 - a^4 \) to its expanded form is:
\[
x^4 - a^4 = (x - a)(x + a)(x^2 + a^2)
\]