Answers

2016-01-13T14:22:38+08:00
x3 + y3 = (x + y)(x2 − xy + y2) [Sum of two cubes]x3 − y3 = (x − y)(x2 + xy + y2) [Difference of 2 cubes]Where do these come from? If you multiply out the right side of each, you'll get the left side of the equation.Note: We cannot factor the right hand sides any further.We use the above formulas to factor expressions involving cubes, as in the following example.ExampleFactor \displaystyle{64}{x}^{3}+{125}64x​3​​+125Answer:We use the Sum of 2 Cubes formula given above.64x3 + 125= (4x)3 + (5)3= (4x + 5)[(4x)2 − (4x)(5) + (5)2]= (4x + 5)(16x2 − 20x + 25)As mentioned above, we cannot factor the expression in the second bracket any further. It looks like it could be factored to give \displaystyle{\left({4}{x}-{5}\right)}^{2}(4x−5)​2​​, however, when we expand this it gives:\displaystyle{\left({4}{x}-{5}\right)}^{2}={16}{x}^{2}-{40}{x}+{25}(4x−5)​2​​=16x​2​​40x+25This "perfect square trinomial" is not the same as the expression we obtained when factoring the sum of 2 cubes.ExercisesFactor:(1) x3 + 27Answer(2) 3m3 − 81
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