Question

A quadratic polynomial is exactly divisible by (x + 1) & (x + 2) and leaves the remainder 4 after division by (x + 3) then that polynomial is

• A. $x^2+6x+4$
• B. $2x^2+6x+4$
• C. $2x^2+6x-4$
• D. $x^2+6x-4$

Answer 1

Answer: (B)

$2x^2+6x+4$

Let quadratic polynomial
$p\left(x\right)=a{x}^2+bx+c$
Now, it is exactly divisible by x + 1 and x + 2
So, p(-1) and p(-2) will be equal to zero.
$p(-1)=0$
$a-b+c=0$ .... (i)
and $p(-2)=0$
$4a-2b+c=0$ ..... (ii)
and when p(x) is divided by x + 3 it leave remainder 4.
$p(-3)=4$
$9a-3b+c=4$ ....(iii)
Subtract (ii) from (i)
$-3a+b=0$
$\Rightarrow b=3a$
put $b=3a$ in equation (iii)
$9a-3\left(3a\right)+c=4$
$\Rightarrow c=4$
By putting the value of c=4 in (i) and (ii).we get
$a-b+4=0$ .... (iv)
$4a-2b+4=0$ .....(v)
Multiplying equation (iv) by 2
2a - 2b + 8 = 0
4a - 2b + 4 = 0
- + -
-----------------------
-2a + 4 = 0
a = 2
When $a=2,b=3a\Rightarrow b=3\times 2=6$
So, quadratic polynomial $=a{x}^2+bx+c$
$=2x^2+6x+4.$