Question

Which pattern can be used to factor the high degree polynomial 9x^10−36x^2?

Answer 1

To factor the polynomial \( 9x^{10} - 36x^2 \), we can start by identifying any common factors in both terms.

1. **Identify the greatest common factor (GCF)**: The GCF of the coefficients \( 9 \) and \( 36 \) is \( 9 \). Additionally, both terms have \( x^2 \) as a common factor. Thus, we can factor out \( 9x^2 \).

2. **Factor out \( 9x^2 \)**:
\[
9x^{10} - 36x^2 = 9x^2(x^8 - 4)
\]

3. **Recognize the difference of squares**: The expression \( x^8 - 4 \) is a difference of squares, since \( 4 \) can be written as \( 2^2 \):
\[
x^8 - 4 = (x^4)^2 - (2)^2
\]
This can be factored using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \):
\[
x^8 - 4 = (x^4 - 2)(x^4 + 2)
\]

4. **Combine the factors**: Putting it all together, we have:
\[
9x^{10} - 36x^2 = 9x^2(x^4 - 2)(x^4 + 2)
\]

Thus, the factored form of the polynomial \( 9x^{10} - 36x^2 \) is:
\[
9x^2(x^4 - 2)(x^4 + 2)
\]