To factor the given polynomial expression [tex]\(\left(x^2-4\right)\left(x^2-9\right)\)[/tex] into linear factors, let's break it down step by step.
1. Identify and factorize each quadratic polynomial separately:
- The first quadratic polynomial is [tex]\(x^2 - 4\)[/tex].
The expression [tex]\(x^2 - 4\)[/tex] is a difference of squares, which can be factored using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]. Here, [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex], thus:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
- The second quadratic polynomial is [tex]\(x^2 - 9\)[/tex].
Similarly, [tex]\(x^2 - 9\)[/tex] is a difference of squares. Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex], so:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
2. Combine the factorizations:
We substitute the factored forms back into the original expression:
[tex]\[
\left(x^2 - 4\right)\left(x^2 - 9\right) = (x - 2)(x + 2) \cdot (x - 3)(x + 3)
\][/tex]
3. Write the final product as linear factors:
Thus, the polynomial [tex]\(\left(x^2-4\right)\left(x^2-9\right)\)[/tex] factored into linear factors is:
[tex]\[
(x - 2)(x + 2)(x - 3)(x + 3)
\][/tex]
Therefore, the product as linear factors is:
[tex]\[
(x - 2)(x + 2)(x - 3)(x + 3)
\][/tex]
