 Hi and welcome to the session. I'm Asha and I'm going to help you solve the following problem that says Factories each of the following So let us first learn some simple identities first is u cube Plus v cube plus 3 u square v Plus 3 u v square v equal to u plus v whole cube and The second one is u cube minus v cube minus 3 u square v Plus 3 u v square is equal to u minus v whole cube So these two identities are key idea With the help of which we will factorize the given five problems Let us now start with the solution and the first one is 8a cube Plus b cube plus 12 a square b Plus 6 a b square now we will write it in the form of u cube plus v cube Plus 3 u square v plus 3 u v square so that it can further be written as u plus v whole cube So let us start now 8a cube can be written as 2 a whole cube and v cube can be written as v whole cube and 12 a square b We need to write it as 3 u square v so 3 use 2a square into v is b and now writing 6 a b square as 3 into a that is 2a into v square so v is b Square now 8 a cube plus b cube plus 12 a square b Plus 6 a b square Square can be further written as whole cube Plus b whole cube 12 a square b can be written as 3 into 2 a square into b Plus 6 a b square can be written as 3 into 2a into b square Now on comparing it with the left hand side of the first identity We find it in the form of u cube plus v cube plus 3 u square v plus 3 u v square Where u is 2a and v is b and That's when applying the first identity. It can be written as 2a plus b whole cube that is 2 a plus b Into 2 a plus b into 2 a plus b And that's on factorizing we have 2 a plus b Into 2 a plus b Into 2 a plus b So this completes the first part and now proceeding on to the second part Where we have 8 a cube Minus b cube minus 12 a square b Plus 6 a b square Now again, we will try to write it in the form of The left hand side of the second identity Which is u cube minus v cube minus 3 u square v plus 3 u v square So that it can further be written as u minus v whole cube starting with 8 a whole cube This can be written as 2 a whole cube b cube can be written as b whole cube 12 a square b can be written as 3 Into u square that is a whole square into b and 6 a b square can be written as 3 Into 2 a into b square And so The given problem 8 a cube Minus b cube minus 12 a square b plus 6 a b square can be written as 2 a whole cube Minus b whole cube minus 3 times of 2 a into 2 a whole square into b plus 3 times of 2 a Into b square Now this is in the form of the left hand side of the second identity which is equal to 2 a minus b whole cube this is By key idea now this can further be written as 2 a minus b into 2 a minus b into 2 a minus b and That's on factorizing the second one begin to answer as 2 a minus b into 2 a minus b into 2 a minus b So this completes the second part And now proceeding on to the third part, which is 27 minus 125 a cube Minus 135 into a Plus 225 a square now here 27 can be written as 3 whole cube 125 a cube can be written as 5 a whole cube 135 a can be written as 3 into 3 square into 5 a and 225 a square can be written as 3 into 3 into 5 a whole square 3 whole cube Minus 125 a cube can be written as 5 a whole cube then minus 135 a can be written as 3 Into 3 square into 5 a and we have plus and 225 a square can be written as 3 into 3 Into 5 a whole square and thus it is in the form of the left hand side of the second identity Who's formula is 3? minus 5 a whole cube And that's it can be further written as 3 minus 5 a Into 3 minus 5 a Into 3 minus 5 a and that's on factorizing our answer is 3 minus 5 a Into 3 minus 5 a Into 3 minus 5 a So this completes the third part and now proceeding on to the fourth part which is 64 a cube minus 27 b cube Minus 144 a square b Plus 108 a b square now 64 a cube can be written as 4 a whole cube 27 b cube can be written as 3 b whole cube 144 a square b can be written as 3 Into 4 a whole square into 3 b and 108 a b square can be written as 3 into 4 a Into 9 b whole square and Thus the given polynomial can be further written as 4 a whole cube Minus 3 b whole cube minus 3 Into 4 a square into 3 b Plus 3 into 4 a Into 9 b whole square sorry, this is 3 b this is also 3 b That's 108 can be written as 3 into 4 a into 3 b whole square Now this is further equal to 4 a minus 3 b Whole cube and this is by a key idea To this is 4 a and y is 3 b. So this is further equal to 4 a minus 3 b Into 4 a minus 3 b into 4 a minus 3 b as I'm factorizing The fourth part we get an answer as 4 a minus 3 b Into 4 a minus 3 b Into 4 a minus 3 b which completes the fourth part and now proceeding on to the last one Where we have to factorize them 27 p q minus 1 upon 216 minus 9 upon 2 p square Plus 1 upon 4 p of 27 p q can be written as 3 p whole cube and 1 upon 216 can be written as 1 upon 6 Whole cube 9 upon 2 p square can be written as 3 Into 3 b whole square into 1 upon 6 and 1 upon 4 p can be written as 3 Into 3 p into 1 upon 6 whole square and thus the given polynomial Can be written as 3 p whole cube minus 1 upon 6 whole cube minus 3 into 3 p square into 1 upon 6 Plus 3 into 3 p Into 1 upon 6 whole square which is in the form of the left-hand side of the second identity where x is equal to 3 p and Y is equal to 1 upon 6 and that's when applying the identity it can further be written as 3 p minus 1 upon 6 whole cube Which is equal to 3 p minus 1 upon 6 Into 3 p minus 1 upon 6 Into 3 p minus 1 upon 6 So this is our answer 3 minus 1 upon 6 3 p minus 1 upon 6 and 3 p minus 1 upon 6 which completes the last part an Instalization so hope you enjoyed it take care and do remember the formulas while doing these types of problem have a good day