(a) To find the function q(x) q(x) q(x), we need to expand and simplify the expression (x+3)4−3(x+3)2 (x+3)^4 - 3(x+3)^2 (x+3)4−3(x+3)2.
(b) The function r(x) r(x) r(x) is already simplified, so we can directly evaluate it for any given x x x.
(c) The function s(x) s(x) s(x) is also already simplified, so we can directly evaluate it for any given x x x.
We start with the function q(x)=(x+3)4−3(x+3)2 q(x) = (x + 3)^4 - 3(x + 3)^2 q(x)=(x+3)4−3(x+3)2. By expanding and simplifying, we find: q(x)=(x+3)2((x+3)2−3) q(x) = (x + 3)^2 \left( (x + 3)^2 - 3 \right) q(x)=(x+3)2((x+3)2−3) Thus, the simplified form of q(x) q(x) q(x) is: q(x)=(x+3)2(x2+6x+6) q(x) = (x + 3)^2 \left( x^2 + 6x + 6 \right) q(x)=(x+3)2(x2+6x+6)
The function r(x) r(x) r(x) is given as: r(x)=x4−3x2−2 r(x) = x^4 - 3x^2 - 2 r(x)=x4−3x2−2 This expression is already in its simplest form.
The function s(x) s(x) s(x) is given as: s(x)=x4256−3x216 s(x) = \frac{x^4}{256} - \frac{3x^2}{16} s(x)=256x4−163x2 This expression is also in its simplest form.
The simplified forms of the functions are:
Thus, the final answers are: q(x)=(x+3)2((x+3)2−3),r(x)=x4−3x2−2,s(x)=x4256−3x216 \boxed{q(x) = (x + 3)^2 \left( (x + 3)^2 - 3 \right), \quad r(x) = x^4 - 3x^2 - 2, \quad s(x) = \frac{x^4}{256} - \frac{3x^2}{16}} q(x)=(x+3)2((x+3)2−3),r(x)=x4−3x2−2,s(x)=256x4−163x2
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