 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/1/a/solution

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2}) .

Expanding the right side cancels the two mixed terms of each degree. The quadratic factor satisfies

a^{2} + ab + b^{2} = (a + b /2)^{2} + 3 b^{2} /4

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