1. **Stating the problem:** Find the roots of the polynomial $$x^4 + 15x^3 + 70x^2 + 120x + 64$$ using the grouping method (GP). 2. **Recall the grouping method:** Group terms to factor the polynomial step-by-step. The goal is to rewrite the polynomial as a product of two quadratic polynomials. 3. **Group terms:** Group the polynomial as $$\left(x^4 + 15x^3 + 70x^2\right) + \left(120x + 64\right)$$. 4. **Factor each group:** - From the first group, factor out $$x^2$$: $$x^2(x^2 + 15x + 70)$$. - From the second group, factor out 8: $$8(15x + 8)$$. 5. **Check for common binomial factors:** The terms inside parentheses are $$x^2 + 15x + 70$$ and $$15x + 8$$, which are not the same, so grouping as is does not help. 6. **Try to factor the polynomial as a product of two quadratics:** Assume $$x^4 + 15x^3 + 70x^2 + 120x + 64 = (x^2 + ax + b)(x^2 + cx + d)$$. 7. **Expand the right side:** $$ (x^2 + ax + b)(x^2 + cx + d) = x^4 + (a + c)x^3 + (ac + b + d)x^2 + (ad + bc)x + bd $$ 8. **Match coefficients:** - Coefficient of $$x^3$$: $$a + c = 15$$ - Coefficient of $$x^2$$: $$ac + b + d = 70$$ - Coefficient of $$x$$: $$ad + bc = 120$$ - Constant term: $$bd = 64$$ 9. **Find integer pairs for $$b$$ and $$d$$ such that $$bd = 64$$:** Possible pairs: (1,64), (2,32), (4,16), (8,8), and their negatives. 10. **Try $$b = 8$$ and $$d = 8$$:** Then $$bd = 64$$. 11. **Use $$a + c = 15$$ and $$ac + b + d = 70$$:** - $$ac + 8 + 8 = 70 \Rightarrow ac = 54$$ 12. **Use $$ad + bc = 120$$:** - $$a \cdot 8 + b \cdot c = 8a + 8c = 8(a + c) = 8 \times 15 = 120$$ which matches perfectly. 13. **Solve for $$a$$ and $$c$$:** - $$a + c = 15$$ - $$ac = 54$$ 14. **Find $$a$$ and $$c$$ as roots of $$t^2 - 15t + 54 = 0$$:** - Discriminant $$\Delta = 15^2 - 4 \times 54 = 225 - 216 = 9$$ - Roots: $$t = \frac{15 \pm 3}{2}$$ - So $$t = 9$$ or $$t = 6$$ 15. **Therefore, $$a = 9$$ and $$c = 6$$ (or vice versa).** 16. **Write the factorization:** $$x^4 + 15x^3 + 70x^2 + 120x + 64 = (x^2 + 9x + 8)(x^2 + 6x + 8)$$ 17. **Factor each quadratic further:** - $$x^2 + 9x + 8 = (x + 1)(x + 8)$$ - $$x^2 + 6x + 8$$ cannot be factored easily with integers, so use quadratic formula: $$x = \frac{-6 \pm \sqrt{36 - 32}}{2} = \frac{-6 \pm 2}{2}$$ - Roots: $$x = -2$$ or $$x = -4$$ 18. **Final roots:** $$x = -1, -8, -2, -4$$ **Answer:** The roots of the polynomial are $$\boxed{-1, -8, -2, -4}$$.