This question has been studied a bit because it is actually of practical relevance to computer algebra. The tightest known bounds are surprisingly large. A good review is Bounds on factors in $\mathbb{Z}[x]$, John Abbott, which according to his CV has been accepted for publication by the Journal of Symbolic Computation. His paper also contains a number of examples and an argument for why the bounds must necessarily be larger than you might hope.

When I wrote this answer there was another one with a couple of small examples, but it's been deleted, so I'll add some small examples in different categories from Abbott's paper.

Large height reducible factor of $x^d-1$: $$\begin{eqnarray}x^{12} - 1 &amp; = &amp; (x-1)(x^2+1)(x^2-x+1)\; (x+1)(x^2+x+1)(x^4-x^2+1)\\
&amp; = &amp; (x^5 - 2x^4 + 3x^3 - 3x^2 + 2x - 1)(x^7 + 2x^6 + x^5 - x^4 - x^3 + x^2 + 2x + 1)\end{eqnarray}$$

Large height irreducible factor of polynomial of height $1$: $$x^4 + x^3 - x^2 - 1 = (x-1) (x^3 + 2x^2 + x + 1)$$

Large height reducible factor of polynomial of height $1$: $$x^3 + x^2 - x - 1 = (x-1)(x+1)^2 = (x-1)(x^2 + 2x + 1)$$

Maximum absolute value of polynomial coefficients
I have a  7 different columns with various attributes &amp; 500 rows with corresponding values for these attributes [Numbers,texts,YES/No etc]..Example only 2 column with values shown.

1st column have range &amp; rest 6 have values need to be counted based on the ranges.

Column 1 is called level ,  I have levels from 1 to 7....this will repeat for next groups.

Level 1 is main level &amp; remaining levels [2 to 7] are sub levels. Need to find missing value counts for this group and subsequent groups.

PS: this is for a Bill OF material [BOM]

example

   Level Number Description
-   1   6586    ABC
    2   6579    XYZ
    3   6689    
    3   7854    123
    4   6011    GHF
G1  4   OOPO    YUI
    4   5589    OIK
    5   2132    
    6   4178    BUY
-   7   7145    CRI
*   1   8245    WES
    2   6666    RED
    3   1025    TRY
G2  3   9898    UIO
    4   4567    POL
    5   1234    WIP
    6   987     III
*   7   7787    RTE
+   1   6652    WED
    2   5425    
    3   9899    TRY
G3  3   6452    OOP
    4   3452    POE
    5   7890    
    6           LLK
+   7   8889    RET


Result expected [Aim is to find count of missing/wrong values]

Main ITEM      Number Description
    G1           1       3     [1 text in number column,2 blank/1 number in Dsecr column]
    G2           0       0     [ 0 error , as both columns filled proeprly]
    G3           1       2     [1 blank in Number column,2 blanks in Descr. column]


   Name &amp; description corresponding to level 1 are always constant.


To make things bit more clear...

If i use MATCH function with lookupvalue ABC it will give position 3 &amp; MATCH function for WES will give posiiton 9. Based on this range ,Row 3 to row 8,[Row 9 will start of anotther range] need count of blanks or texts for NUMBER,Count of blanks for DESCRIPTION and some other attributes which i have not included here.

ABC is a group with 2 to 7 sub levels

    1      6586     ABC
    2      6579     XYZ
    3      6689       
    3      7854     123
    4      6011     GHF
    4      OOPO     YUI
    4      5589     OIK
    5      2132     
    6      4178     BUY
    7      7145     CRI 


WES is a group with next 2 to 7 sub levels

1      8245     WES
2      6666     RED
3      1025     TRY 
3      9898     UIO
4      4567     POL
5      1234     WIP
6      0987     III
7      7787     RTE


WED is next group

    1      6652     WED
    2      5425     
    3      9899    TRY
    3      6452     OOP
    4      3452     POE
    5      7890     
    6               LLK
    7      8889     RET

