1. The problem asks to factor the given polynomial expressions. 2. We use the formula for the cube of a binomial: $$ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 $$ 3. For each polynomial, we identify terms matching this pattern and rewrite them as a cube. 4. Problem 413: $$ a^3 - 3a^2 + 3a - 1 $$ Recognize this as $$ (a - 1)^3 = a^3 - 3a^2 + 3a - 1 $$ 5. Problem 414: $$ x^3 - 9x^2 + 27x - 27 $$ Recognize this as $$ (x - 3)^3 = x^3 - 9x^2 + 27x - 27 $$ 6. Problem 415: $$ 8x^3 + 36x^2y + 54xy^2 + 27y^3 $$ Recognize this as $$ (2x + 3y)^3 = 8x^3 + 36x^2y + 54xy^2 + 27y^3 $$ 7. Problem 416: $$ a^3 - 6a^2b + 12ab^2 - 8b^3 $$ Recognize this as $$ (a - 2b)^3 = a^3 - 6a^2b + 12ab^2 - 8b^3 $$ 8. Problem 417: $$ -\frac{1}{27} x^3 - \frac{9}{4} xy^4 + \frac{1}{2} x^2y^2 + \frac{27}{8} y^6 $$ This is more complex; rearranged and factored as $$ -\left( \frac{1}{3} x + \frac{3}{2} y^2 \right)^3 $$ 9. Problem 418: $$ m^9 - 12m^6n^2 + 48m^3n^4 - 64n^6 $$ Recognize this as $$ (m^3 - 4n^2)^3 = m^9 - 12m^6n^2 + 48m^3n^4 - 64n^6 $$ 10. For 425a: $$ -27 + a^9 - 9a^6 + 27a^3 $$ Rearranged as $$ a^9 - 9a^6 + 27a^3 - 27 $$ Recognize as $$ (a^3 - 3)^3 $$ 11. For 425b: Complete the cube: $$ \left( \frac{1}{2} a + \frac{3}{2} b \right)^3 = \frac{1}{8} a^3 + \frac{3}{4} a^2 b + \frac{9}{4} a b^2 + \frac{27}{8} b^3 $$ Final answers: - 413: $$(a - 1)^3$$ - 414: $$(x - 3)^3$$ - 415: $$(2x + 3y)^3$$ - 416: $$(a - 2b)^3$$ - 417: $$-\left( \frac{1}{3} x + \frac{3}{2} y^2 \right)^3$$ - 418: $$(m^3 - 4n^2)^3$$ - 425a: $$(a^3 - 3)^3$$ - 425b: $$\left( \frac{1}{2} a + \frac{3}{2} b \right)^3$$