Question

Which of the following expressions is equivalent to (y-3)^4 ?

Answer 1

ANSWER :

The answer is A. y^4 - 12y^3 + 54y^2 - 108y + 81

EXPLANATION :

From the problem, we have the expression :

$(y-3)^4$

we can write it as :

$(y-3)^4=[(y-3)^2]^2$

Perform the first power to the 2nd using :

$(a^2\pm b^2)=a^2\pm2ab+b^2$

$[(y-3)^2]^2=(y^2-6y+9)^2$

Next is to perform the next square :

$(y^2-6y+9)^2=(y^2-6y+9)(y^2-6y+9)$

We can distribute each term of the first parenthesis to the second parenthesis as :

$(a+b+c)(a+b+c)=a(a+b+c)+b(a+b+c)+c(a+b+c)$

Then :

$\begin{gathered} (y^2-6y+9)(y^2-6y+9) \\ =y^2(y^2-6y+9)-6y(y^2-6y+9)+9(y^2-6y+9) \\ \text{ Then simplify :} \\ =(y^4-6y^3+9y^2)-(6y^3-36y^2+54y)+(9y^2-54y+81) \\ \text{ Combine like terms :} \\ =y^4-6y^3-6y^3+9y^2+36y^2+9y^2-54y-54y+81 \\ =y^4-12y^3+54y^2-108y+81 \end{gathered}$