Question

Find value of a & b so that the polynomial x^3 -10x^2 +ax +b is exactly divisible by (x-1) as well as (x-2)​

Answer 1

Step-by-step explanation:

We have given;

The equation which needs to be solved i.e,

x^3 -10x^2 +ax +b

and its factors:

(x-1) and (x-2)

Now we know how to derive zeroes of any polynomial from its factors.

Hence the zeroes of the polynomial are:

= x-1 =0

= x = 1;

and, = x-2 =0

= x = 2;

So, if we put the zeros of polynomial one by one in the polynomial then we will get zero as its result.

So, lets put it.

When, x = 1;

The equation formed will be;

= (1)^3 -10(1)^2 +a(1) +b = 0;

= 1 - 10 + a + b = 0;

= -9 + a + b = 0;

= a + b = 9;

Now, when x = 2

The equation formed will be;

= (2)^3 -10(2)^2 +a(2) +b = 0;

= 8 - 40 + 2a + b = 0;

= -32 + 2a + b = 0;

= 2a + b = 32;

Now, we have two equations and i.e,

= a + b = 9; (i)

= 2a + b = 32; (ii)

{ Here, I will use elimination method but one can use any of the methods like substitution method, etc. to solve such problem. }

So, for elimination, I will subtract equation (i) from (ii).

Thus, = (2a + b) - (a+b) = 32 - 9;

= 2a + b - a - b = 23;

= a = 23. (here b got eliminate) (iii)

Now, b equals;

= a + b = 9; (using (i) and (iii))

= 23 + b = 9;

= b = 9 -23;

= b = - 14;

Hence, value of a = 23 and b = -14.

That's all.