Factor Polynomial De6472

1. **State the problem:** Factor the polynomial $$x^4 - 7x^2 - 18$$ completely over the integers. 2. **Recognize the structure:** Notice that the polynomial is in terms of $x^4$ and $x^2$. Let us use substitution to simplify. 3. **Substitution:** Let $$y = x^2$$. Then the polynomial becomes: $$y^2 - 7y - 18$$ 4. **Factor the quadratic in $y$:** We look for two numbers that multiply to $$-18$$ and add to $$-7$$. These numbers are $$-9$$ and $$2$$. 5. **Write the factorization:** $$y^2 - 7y - 18 = (y - 9)(y + 2)$$ 6. **Back-substitute $y = x^2$:** $$(x^2 - 9)(x^2 + 2)$$ 7. **Factor further if possible:** $$x^2 - 9$$ is a difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$ 8. **Final factorization:** $$(x - 3)(x + 3)(x^2 + 2)$$ 9. **Check for irreducibility:** The quadratic $$x^2 + 2$$ cannot be factored further over the integers. **Answer:** $$(x - 3)(x + 3)(x^2 + 2)$$