Question

Which of the following is not a factor of $(a+b–c{)}^3–[a^3+b^3–c^3]$?

• A. a + b
• B. b - c
• C. a + b - c
• D. a - c

Answer 1

The correct option is C a + b - c

$(a+b–c{)}^3–[a^3+b^3–c^3]$
$=(a+b–c{)}^3+c^3-(a^3+b^3)$
$=(a+b+c+c)\left[\right(a+b+c{)}^2-(a+b-c)c+c^2]-(a+b\left)\right(a^2-ab+b^2)$
[$\therefore x^3+y^3=(x+y)(x^2-xy+x{y}^2)$]
$=(a+b)[a^2+b^2+c^2+2ab-2bc-2ca-ca-bc+c^2+c^2-a^2+ab-b^2]$ [$\therefore (a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$]
$=(a+b)[3c^2+3ab-3bc-3ca]$
$=(a+b)[3c^2-3ca+3ab-3bc]$
$=(a+b)\left[3c\right(c-a)-3b(c-a\left)\right]$
$=(a+b)(c-a)(3c-3b)$
$=3(a+b)(c-a)(c-b)=3(a+b)(b-c)(a-c)$
Thus, a + b – c is not a factor.
Hence, the correct answer is option (3).