in this video we're going to focus on factoring sums and difference of Cubes so let's say if we want to factor the expression X Cub + 8 now there is an equation that you want to use and here it is a the 3 + b the 3 this is equal to a + b time uh a 2 minus AB plus b^2 so you need to realize that a to the 3 is the same as X cub in this problem therefore if you take the cube root of both sides a is equal to X now B the 3 is equal to 8 and the cube root of 8 is two so B is equal to two and now we just got to plug in everything into the formula so just keep in mind a is X B is 2 so A + B that's going to be x + 2 a 2 is x^2 a * b or x * 2 that's 2X and b^2 is 2^ 2 2 * 2 is 4 and so that's how you can Factor uh this particular expression but let's go ahead and try another example let's say if we want to factor the expression X Cub + 125 feel free to pause the video and try this example so notice that a to the 3 is the same as X Cub for this problem therefore a is X and B the 3 is 125 now what is the cube root of 125 what times what * 1 is 125 the answer is five so instead of writing a plus b this is going to be x + 5 and then it's multiplied by A2 or x^2 minus a * b or 5 * X Plus b^2 which is 5^ 2 and so that's how you can factor a sum of cubes now let's try and another example let's say if it's 8 x 3r + 27 try that example so we can see that a the 3 is equal to 8 x Cub so if that's the case what is the value of a the cube root of 8 is 2 and the cube root of x 3r is simply X so a is 2x now B Cub is 27 and we need to take the cube root of 27 to find B that means B is equal to 3 so now using the formula A + B it's going to be 2x + 3 * a^ 2 which is 2x^2 or 2x * 2X and that's 4x^2 minus a which is 2 2 x * 3 and that's 6 x + b^ 2 which is 3^ 2 or 9 so that's the answer so let's try one more example with sum of cubes try this one 25 x Cub + 64 YB so we can see that a cub is 25x cub and B Cub is 64 y Cub actually I can't use 25 let's take out 25 and let's use uh 27 instead 25 is not a perfect Cube so a is going to be the cube root of 27 which is three and the cube root of x Cub is X B is going to be the cube root of 64 and the cubot of Y cube is y so now that we have a and b we could find the answers so now let's substitute a is 3x and B is 4 Y and then it's a 2 or 3x^ 2us a B which is uh 3x * 4 y + B ^2 or 4 y^ 2 so this is equal to 3x + 4 y now 3x^2 that's 3x * 3x which is going to be 9 x^2 -3x * 4 y it's going to bexy and 4 y^2 4^ 2 is 16 so this is going to be plus 16 y^2 so this is the answer now the next equation that you need to be familiar with is the difference of Cubes so a cub minus BB and this is equal to a minus B * a 2 + AB plus b s so if there's a negative sign between a cub and B Cub there going to be a negative sign between a and b and then this sign is going to change to a positive in the last example we had this equation A 3r + B 3r is equal to a + b * a 2 minus a + b 2 so if you want to come up with a a generalized formula for both here it is this is going to be plus and then minus actually let me put it in different colors so it's going to be a plus or minus and then a s the sign is going to flip at this point so I'm going to put the the blue one on top but it's going to be minus the red one on the bottom that's going to be plus AB + B2 so as you can see the sign is going to stay the same at first and then it's going to reverse now let's try some examples try this one X Cub - 216 so a is equal to X and B is the cube root of 216 which is 6 so this is going to be x - 6 * a^ 2 or x^2 + a or + 6 * x + b^2 which is 6^ 2 or 36 and that's it for that example let's try some more try this one 64 y Cub minus 125 so a cub is 64 YB B 3 is25 the cube root of 125 is 5 the cube root of 64 is four so a is going to be 4 y so a minus B that's 4 y - 5 * a^ 2 which is 4 y^ 2 or 4 y * 4 Y which is 16 y^ 2 plus a or 4 y * 5 which is uh 20 y + b^ 2 or 5^ 2 which is 25 try this one 8 y cubus 27 so a is or a cub is 8 y Cub B to the 3 is 27 7 so a is going to be 2 Y and B is equal to 3 so it's going to be a - b or 2 y - 3 * a^ 2 which is 2 y * 2 Y and that's 4 y^ 2+ a 2 y * 3 is 6 y plus b^ 2 or 3^ 2 which is n so as you can see these problems are not too difficult to do but let's try some different examples try this one x 6 - 64 y 9th so a cub is X to the 6 power and B Cub is 64 y 9th so what is the cube root of x 6 to find a cube root you can raise both sides to the 1/3 power so basically you're dividing 6 by 3 6id 3 is 2 so a is X2 now what about B the cube root of 64 is 4 the cube root of Y the 9 is basically 9 / 3 so it's going to be y the 3r so a minus B that's x^2 - 4 y Cub now what's a s so if a is x s a 2 is going to be X to 4th now what about AB so multiply X2 and 4 y Cub that's going to be positive 4 x^2 y Cub plus b^2 now B is 4 YB so b^ 2 is 4 * 4 which is 16 and YB * YB is y 6 so it's + 16 y 6 that's the answer