Factor Polynomial 3Bc31A

1. **State the problem:** Factor the polynomial $$x^4 - 7x^2 - 18$$ completely over the integers. 2. **Rewrite the polynomial:** Notice that the polynomial is in terms of $x^4$ and $x^2$. Let $y = x^2$. Then the polynomial becomes: $$y^2 - 7y - 18$$ 3. **Factor the quadratic in $y$:** We want two numbers that multiply to $-18$ and add to $-7$. These numbers are $-9$ and $2$ because: $$-9 \times 2 = -18$$ $$-9 + 2 = -7$$ 4. **Write the factorization in $y$:** $$y^2 - 7y - 18 = (y - 9)(y + 2)$$ 5. **Substitute back $y = x^2$:** $$(x^2 - 9)(x^2 + 2)$$ 6. **Factor further if possible:** - $x^2 - 9$ is a difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$ - $x^2 + 2$ cannot be factored over the integers. 7. **Final factorization:** $$(x - 3)(x + 3)(x^2 + 2)$$