To find the linear factorization of the function f(x) = x4 – 9x2, we can start by recognizing that this is a polynomial function. The first step is to factor out the common term from both parts of the polynomial.
1. **Factor out the common term:**
We can factor out x2 from the function:
f(x) = x2(x2 - 9)2. **Recognize a difference of squares:**
The expression (x2 – 9) is a difference of squares, which can be factored further:
x2 - 9 = (x - 3)(x + 3)3. **Combine the factors:**
Putting it all together, we can express f(x) as:
f(x) = x2(x - 3)(x + 3)4. **Further factorization:**
The factor of x2 can also be expressed as:
x2 = (x)(x)So, the complete linear factorization of the function is:
f(x) = (x)(x)(x - 3)(x + 3)5. **Final result:**
Therefore, the linear factorization of f(x) = x4 – 9x2 is:
f(x) = x2(x - 3)(x + 3)or simply:
f(x) = (x)(x)(x - 3)(x + 3)This is the complete linear factorization, which can be helpful for solving equations or analyzing the behavior of the polynomial.