00:01
Hello friends.
00:01
So in this question one function is given that is f of x is equal to a x cube minus x square plus b x minus 24 equal to 0 and two factors of this equations are given as x minus 2 and x plus 4.
00:17
We need to find out the third factor.
00:20
So let us first put the values in this equation and we will get the two linear equation in the form of a and b.
00:28
So if we put x is equal to 2 we get to 2, we get one linear equation that is 8a plus 2b equal to 28 this is equation 1 and if we put x is equal to minus 4 in this equation we will get the another linear equation that is 64a plus 4b equal to minus 40 and this is equation 2 if we solve equation 1 and 2 we will get the values of a and b as solving 1 and 2 we get a is equal to minus 2 and that of the equation 1 and 2 and that of b is equal to 22.
01:03
Now put this a and b in f of x to get f of x as 2x cube minus 2x cube minus square plus 22x minus 24.
01:19
Now in order to find out the third factor we will use the synthetic division so we will divide this polynomial with given factor x minus 2.
01:26
So minus 2 x cube minus x square plus 22 x 2x.
01:32
Minus 24.
01:33
Here minus 2x cube so we will multiply with minus 2x square we will get minus 2x cube plus 4x square.
01:42
So this two will get cancelled and here we will get minus 5x square plus this as it is 22x minus minus 24.
01:51
Again we need to multiply with minus 5x so we'll write minus 5x here and multiplication will come minus 5x square and minus 5x into minus 2 that is plus 10x.
02:02
After changing the sign this two terms will get cancelled and here it is minus 10x and here it is 22 so we will get here 12x minus 24 so again we need to multiply it with plus 12 and it will become 12x minus 24 so again changing the signs here we will get this as minus this as plus both of them will get cancelled and we get zero here now we have got f of x as x as x minus 2 and minus 2 x square minus 5x plus 12.
02:38
Now this result we have got after dividing with first factor x minus 2.
02:44
Now we will divide the polynomial with another factor that is x plus 4...