Question

If (x – 1) and (x + 2) are two factors of the polynomial $2x^3+m{x}^2–x–n$, then the value of $(m^2–n^2)$ is

• A. -15
• B. 9
• C. -11
• D. 24

Answer 1

The correct option is C -11

Let $f\left(x\right)=2x^3+m{x}^2–x–n$
Also, (x – 1) and (x + 2) are factors of f(x).
By factor theorem, f(1) = 0 and f(–2) = 0.
$\therefore f\left(1\right)=2(1{)}^3+m(1{)}^2–1–n=0$
$\Rightarrow 2+m–1–n=0$
$\Rightarrow m–n+1=0$ …..(i)
Also, f(–2) = $2(–2{)}^3+m(–2{)}^2–(–2)–n=0$
$\Rightarrow –16+4m+2–n=0$
$\Rightarrow 4m–n–14=0$ …..(ii)
Subtracting (i) and (ii), we get:
$(4m–n–14)–(m–n+1)=0$
$\Rightarrow 3m–15=0$
$\Rightarrow m=5$
Substituting m = 5 in (i), we get:
$5–n+1=0$
$\Rightarrow n=6$
$\therefore m^2–n^2=(5{)}^2–(6{)}^2$
$=25–36=–11$
Hence, the correct answer is option (3).