Question

Given that x+2 and x-3 are two factors of x^3 +ax + b, calculate the values of a and b and find the remaining factor.

Answer 1

"here is your solution ,its 3 pages "

" hope helps "

Answer 2

Answer:

The value of a = -7

The value of b = -6

Step-by-step explanation:

Given:-

$p(x)$ = $x^3+ax+b$

Consider Factor 1 as:- $x+2$

Consider Factor 2 as:- $x-3$

So here, we need to find the values of $a , b$

Process:-

Factor 1:-                              Factor 2:-

Take $x+2=0$                     Take $x-3=0$

So, $x=-2$                            So, $x=3$

The Remainder:- $p(-2)$        The Remainder:- $p(3)$

$p(x) = x^3+ax+b\\p(-2) = (-2)^3+a(-2)+b\\p(-2) = -8+(-2a)+b\\p(-2) = -8-2a+b$     $p(x) = x^3+ax+b\\p(3) = (3)^3+a(3)+b\\p(3) = 27+3a+b$

To find $a$, equate Factor 1's $p(x)$ and Factors 2's $p(x)$ equations

So, ⇒ $-8-2a+b=27+3a+b$

     ⇒ $-8-27=3a+2a$

     ⇒ $5a=35$

     ⇒ $a = -35/5$

     ⇒ $a=-7$

Keeping the 'a' value in Factor 1's equation, we get 'b' value

So, $-8=2(-7)-b$

     $-8+14=-b$

     $-6=b$

Thus, a = -7

         b = -6