Questions: (a) q(x)=(x+3)^4-3(x+3)^2 (b) r(x)=x^4-3x^2-2 (c) s(x)=x^4/256-3x^2/16

(a) q(x)=(x+3)^4-3(x+3)^2
(b) r(x)=x^4-3x^2-2
(c) s(x)=x^4/256-3x^2/16
Transcript text: (a) $q(x)=(x+3)^{4}-3(x+3)^{2}$ (b) $r(x)=x^{4}-3 x^{2}-2$ (c) $s(x)=\frac{x^{4}}{256}-\frac{3 x^{2}}{16}$
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Solution

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Solution Steps

Solution Approach

(a) To find the function q(x) q(x) , we need to expand and simplify the expression (x+3)43(x+3)2 (x+3)^4 - 3(x+3)^2 .

(b) The function r(x) r(x) is already simplified, so we can directly evaluate it for any given x x .

(c) The function s(x) s(x) is also already simplified, so we can directly evaluate it for any given x x .

Step 1: Simplifying q(x) q(x)

We start with the function q(x)=(x+3)43(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)23) q(x) = (x + 3)^2 \left( (x + 3)^2 - 3 \right) Thus, the simplified form of q(x) q(x) is: q(x)=(x+3)2(x2+6x+6) q(x) = (x + 3)^2 \left( x^2 + 6x + 6 \right)

Step 2: Expressing r(x) r(x)

The function r(x) r(x) is given as: r(x)=x43x22 r(x) = x^4 - 3x^2 - 2 This expression is already in its simplest form.

Step 3: Expressing s(x) s(x)

The function s(x) s(x) is given as: s(x)=x42563x216 s(x) = \frac{x^4}{256} - \frac{3x^2}{16} This expression is also in its simplest form.

Final Answer

The simplified forms of the functions are:

  • q(x)=(x+3)2((x+3)23) q(x) = (x + 3)^2 \left( (x + 3)^2 - 3 \right)
  • r(x)=x43x22 r(x) = x^4 - 3x^2 - 2
  • s(x)=x42563x216 s(x) = \frac{x^4}{256} - \frac{3x^2}{16}

Thus, the final answers are: q(x)=(x+3)2((x+3)23),r(x)=x43x22,s(x)=x42563x216 \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}}

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