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# Factoring Theorem of polynomials

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A certain type of candle is packaged in boxes that measure 42 cm by 17 cm by 10 cm. The candle company that produced the above packaging has now designed shorter candles. A smaller box will be created by decreasing each dimension of the larger box by the same length. The volume of the smaller box will be at the most 2220 cm3 . What are the maximum dimensions of the smaller box? Solve this problem algebraically. Please use The Factor Theorem.

May 17, 2021

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Let the  decrease in each dimension  =  x

So....we  have  that

(42 - x)  ( 17 - x)  (10 - x)  ≤  2220   simplify

-x^3 + 69x^2 -1304x + 7140 ≤  2220

-x^3  +  69x^2   -1304x  + 4920 ≤  0     mutiply  through  by  -1  and  reverse the inequality sign

x^3  - 69x^2  + 1304x  - 4920  ≥  0        (1)

Note that  one possible  root  that  makes the  left  side  = 0   is wnen x = 5

So....using  sunthetic divsion,  we can    find  the  remaining  polynomial

5  [  1      -69        1304     -   4920   ]

5         -320          4920

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1      -64         984             0

The remaining poynomial   is   x^2    -  64x   + 984

Setting this  to  0   and  solving  for x  gives  the  other  two  roots of  x  ≈ 25.675 or  x ≈ 38.3246

(1)  will be true  on these intervals

[5, 25.6754]   and  [ 38.3246 , inf )

So....when  x  = 5   the max  dimensions  of  the   box are  37, 12  and  5  cm   May 17, 2021