 Quandaries and Queries Who is asking: Student Level: Secondary Question: Factor completely: 3x3 - 24y3 54x6 + 16y3 16xy - 4x - 4y - 1 0.09x2 - 0.16y2 Hi, The essential of factoring is recognizing a pattern. Three of the main patterns you need to recognize are a difference of squares a2 - b2 = (a - b)(a + b) a difference of cubes a3 - b3 = (a - b)(a2 + ab + b2) , and a sum of cubes a3 + b3 = (a + b)(a2 - ab + b2) To be able to apply these facts you need to be able to look at an algebraic expression and see something squared minus something squared or something cubed minus something cubed or something cubed plus something cubed For example in your first problem 3x3 - 24y3 there is a common factor of 3 so it can be written 3x3 - 24y3 = 3(x3 - 8y3). Now I see a difference of cubes since 8 is 23 and hence x3 - 23y3 = x3 - (2y)3 Now factor as a difference of cubes. Problem 2 is similar. Take out a common factor and you will have a sum of cubes since x6 = (x2)3 Problem 4 is similar. In problem 3, 16xy - 4x = 4x(4y - 1) so 16xy - 4x - 4y - 1 = 4x(4y - 1) - 4y -1 In this expression 4y - 1 is almost a common factor. If the expression were 4x(4y - 1) + 4y -1 then 4y - 1 would be a common factor. Do you have a typo? Penny Go to Math Central