 Hello and welcome to the session. In this session we are going to discuss the following question which says that Why do some of the given polynomials in simplest form? The first part is b square minus b minus 7 the whole plus 3 minus 2b plus 8b square the whole Second, x minus 12 the whole plus x cube minus 2x plus 12 the whole and the third part is 2 into a raised to the power 4 minus 5a square plus 1 the whole plus 3a cube minus a the whole Now let us start with the solution. Now in the first part of the question we are given two polynomials and we need to find the sum of the two polynomials in the simplest form First we open the brackets and we get b square minus b minus 7 plus 3 minus 2b plus 8b square since the associative and commutative properties hold in polynomials According to commutative property if we are given any two numbers say a and b then a plus b can also be written as b plus 8 And in associative property if we are given a plus b plus c the whole then it is also equal to a plus b the whole plus c And now these commutative and associative properties hold in polynomials thus we combine like terms and we get now we see what are the like terms here And here we see that b square and 8b square are like terms similarly minus b and minus of 2b are like terms also minus 7 and 3 are like terms Which implies that this expression can be written as b square plus 8b square plus minus of b minus of 2b plus minus of 7 plus 3 And simplifying further we get b square plus 8b square is equal to 9b square plus minus of b minus of 2b is equal to minus of 3b plus minus of 7 plus 3 is equal to minus of 4 Which is equal to 9b square plus of minus of 3b is equal to minus of 3b plus of minus of 4 is equal to minus 4 So we get 9b square minus of 3b minus of 4 as the required sum which is the required answer Now the second part of the question is x minus 12 the whole plus x cube minus of 2x plus 12 the whole Now here also we are given two polynomials that is x minus 12 and x cube minus 2x plus 12 And we need to find the sum and for that we first open the brackets and we get x minus 12 plus x cube minus of 2x plus 12 Now we will combine all the like terms and we get x cube plus minus of 2x plus x plus minus of 12 plus 12 and which is equal to x cube plus minus of 2x plus x is equal to minus of x Plus minus 12 plus 12 is 0 So this is equal to x cube plus minus of x that is minus x So we say that x cube minus x is the required sum In the third part of the question we are given 2 into a raised to power 4 minus 5a square plus 1 the whole plus 3a cube minus a the whole And we have to add these two polynomials Now first we will open the brackets and we get 2 into a raised to power 4 minus 5a square plus 1 plus 3 into a cube minus a Now we see that there are no like terms in this expression so we simply write it in descending parts of a that is 2 into a raised to power 5 plus 3a cube minus 5a square minus of a plus 1 Which is the required sum of the two polynomials This completes our session. Hope you enjoyed this session