Question

Expand $(1-x+x^2{)}^4$.

• A. $1-4x+10x^2-16x^3+19x^4-16x^2+10x^6-4x^7+x^8$
• B. $1-4x+10x^2-16x^3+x^4+16x^5+10x^6+4x^7-x^8$
• C. $1-4x-10x^2+16x^3-19x^4+16x^5-10x^6+4x^7+x^8$
• D. $1-4x-10x^2-16x^3+19x^4-16x^5+x^8$

Answer 1

Answer: (A)

Put $1-x=y$, then
$(1-x+x^2{)}^4=(y+x^2{)}^{4}4$
$={}^4{C}_0{y}^4+{}^4{C}_1{y}^3{\left(x^2\right)}^1+{}^4{C}_2{y}^2{\left(x^2\right)}^2+{}^4{C}_{3}y{\left(x^2\right)}^3$
$+{}^4{C}_4{\left(x^2\right)}^4$
$=y^4+4y^3{x}^2+6y^2{x}^4+4y{x}^6+x^8$
$={\left(1-x\right)}^4+4x^2{\left(1-x\right)}^3+6x^4{\left(1-x\right)}^2+4x^6\left(1-x\right)+x^8$
$=1-4x+10x^2-16x^3+19x^4-16x^2+10x^6-4x^7+x^8$