find the values of p,q and r such that, when the polynomial f(x) =x^3+px^2+qx+r is divided by (x+2),(x-1) and (x-3) the remainders are -48, 0 and 2 respectively.Hence factorise f(x) completely
To solve for p, q and r, apply the remainder theorem. To do so, set each divisor equal to zero and solve for x.
For the first divisor,
For the second divisor,
And for the third divisor,
Then, plug-in the values of x to the dividend and set it equal to their corresponding remainder.
For the first divisor, x=-2 and the remainder is -48.
`-40=4p-2q+r` (Let this be EQ1.)
For the second divisor, x=1 and the remainder is 0.
`-1=p+q+r` (Let this be EQ2.)
And for the third divisor, x=3 and the remainder is 2.
`-25=9p+3q+r` (Let this be EQ3.)
Next, apply elimination method. So, subtract EQ2 from EQ1 to eliminate r.
`(-)` `-1=p+q+r `
`-13=p-q` (Let this be EQ4.)
Then, subtract EQ3 from EQ2 to eliminate r again.
`(-)` `-25=9p+3q+r `
`-12=4p+q ` (Let this be EQ5.)
Then, add EQ4 and EQ5 to eliminate q.
Plug-in the value of p to EQ4, to get the value of q.
And, plug-in the values of p and q to EQ2 to get r.
Hence, `f(x)=x^3-5x^2+8x-4` .
To factor, consider the divisor that has a remainder zero which is x-1. Since the remainder is zero, it indicates that the polynomial is divisible by x-1.
Dividing x^3-5x^2+8x-4 by x-1 result to:
Applying the reverse of FOIL, factor x^2-4x+4.
Hence, the factor form of the function is `f(x)=(x-1)(x-2)^2` .