Question

$x-2\text{is a factor of}f\left(x\right)=x^3-a{x}^2-bx+16.\text{When}f\left(x\right)\text{is divided by}(x-3),\text{the remainder is}-5.\text{The value of a and b is:}$

• A. $a=4,b=4$
• B. $a=5,b=2$
• C. $a=9,b=8$
• D. $a=7,b=7$

Answer 1

The correct option is A $a=4,b=4$

$\text{Since}(x-2)\text{is a factor of}f\left(x\right),\text{using factor theorem,}f\left(2\right)=0.$

$2^3-a\times (2{)}^2-b\times 2+16=08-4a-2b+16=02a+b=12...(i)$

$\text{When f(x) is divided by (x - 3), the}\text{remainder is f(3) by remainder}\text{theorem. The remainder is given to}\text{be (-5).}f\left(3\right)=-53^3-a\times 3^2-b\times 3+16=-527-9a-3b+16=-53a+b=16...\left(ii\right)$

$\text{Subtracting equation (i) from}\text{equation (ii), we get:}(3a+b)-(2a+b)=16-12\Rightarrow a=4$

$\text{Substiuting value of a in (i), we get:}2\times 4+b=12\Rightarrow b=4$