How do you simplify (y+2)^3 - (y+3)(y-3) - 2(y-2)^2 - (2y -1 -y^2)^2 - y^2(-y^2 + 5y -3)(y+2)3(y+3)(y3)2(y2)2(2y1y2)2y2(y2+5y3)?

1 Answer
Nov 7, 2017

2y^4 + 10y^3 + 12y^2 - 12y + 162y4+10y3+12y212y+16

Explanation:

First, let's mark each section of our equation in color coding, so we can more easily recognize where each term comes from.

color(red)((y+2)^3) color(blue)(-(y+3)(y-3)) color(green)( - 2(y-2)^2) color(orange)(-(2y-1-y^2)^2) color(pink)(-y^2(-y^2+5y-3))(y+2)3(y+3)(y3)2(y2)2(2y1y2)2y2(y2+5y3)

Next, I'm going to simplify this step-by-step. First, we'll simplify color(red)("red")red. For this, we simply distribute thecolor(white)(.)^3.3to the two terms inside the parenthesis, giving us:

color(red)((y^3+8))(y3+8)

Now we simplify color(blue)("blue")blue by multiplying the yy of the first term by the yy, then -33 of the second term. Then, repeat with the 33 of the first term.

So first, the yy

color(blue)(y * y = y^2)yy=y2
color(blue)(y * (-3) = (-3y))y(3)=(3y)

Now for the 33

color(blue)(3 * y = 3y)3y=3y
color(blue)(3 * (-3) = (-9)3(3)=(9)

Put these in order in a new term, remembering to keep our negative from subtraction, which says:

color(blue)(-(y^2 cancel(- 3y + 3y) - 9)) rArr color(blue)(-(y^2-9))

Now we simplify the color(green)("green"). This is very similar to how we handled color(blue)("blue").

color(green)(-2(y-2)^2) rArr color(green)((-2y+4)^2) rArr color(green)((-2y+4)(-2y+4))
"---"
color(green)((-2y) * (-2y)=4y^2)
color(green)((-2y) * 4=(-8y))
"---"
color(green)(4 * (-2y)=(-8y))
color(green)(4 * 4=16)
Thus, color(green)((4y^2-16y+16))

Now color(orange)("orange"), similar again to color(blue)("blue") and color(green)("green").

color(orange)((-2y-1-y^2)^2) rArr color(orange)((-2y-1-y^2)(-2y-1-y^2))
"---"
color(orange)((-2y) * (-2y) = 4y^2)
color(orange)((-2y) * (-1) = 2y
color(orange)((-2y) * (-y^2) = 2y^3
"---"
color(orange)((-1) * (-2y) = 2y
color(orange)((-1) * (-1) = 1
color(orange)((-1) * (-y^2) = y^2
"---"
color(orange)((-y^2) * (-2y) = 2y^3
color(orange)((-y^2) * (-1) = y^2
color(orange)((-y^2) * (-y^2) = y^4
"---"
color(orange)((4y^2 + 2y + 2y^3+2y+1+y^2+2y^3+y^2+y^4)
Combine like terms to get:
color(orange)((y^4+4y^3+6y^2+4y+1))

And finally, color(pink)("pink").

color(pink)(-y^2(-y^2+5y-3)) rArr color(pink)((y^4-5y^3+3y^2))

Now, put all of your answers from this together, and drop the parenthesis.

color(red)(y^3+8) color(blue)(-y^2-9) color(green)(+ 4y^2 - 16y + 16) color(orange)(+ y^4+4y^3+6y^2+4y+1) color(pink)(+y^4-5y^3+3y^2)

Finally, combine your like terms.

2y^4 + 10y^3 + 12y^2 - 12y + 16