Question

consider two polynomials f(x)= x^{3} -2 x^{2} +1/2 and g(x)= x^{3} -3x+1 if f(x) is divided by x+1 to obtain remainder r_{1} and g(x) is divided by (x+2) to obtain remainder r_{2} .find r_{1} ^2+ r_{2} ^2- r_{1} r_{2}

Answer 1

Hello

P(x) = x³-2x²+1/2

Divided by x+1

So ,.
By remainder theorem

P(-1) = r1
So
(-1)³-2(-1)²+1/2
= - 1-2+1/2
= 1/2-3 = (-5)/2


g(x) = x³-3x+1

Divided by x+2

By remainder theorem
P(-2) = r2

So
(-2)³-3(-2)+1
= - 8+6+1
= - 2+1
= (-1)

Now ,
R1 =-5/2
R2 = (-1)

So,
R1² + R2²-R1R2
$( { \frac{ - 5}{2}) }^{2} + {( - 1)}^{2} - ( - 1)( \frac{ - 5}{2} ) \\ = \frac{25}{4} + 1 - \frac{5}{2} \\ \\ = \frac{25 + 4 - 10}{4} \\ \\ = \frac{19}{4}$


Hope this will be helping you ✌️