The <em><u>correct answer</u></em> is:
3 inches.
Explanation:
We will use the Rational Roots Theorem to solve this.
First we can divide both sides of the equation by 4 in order to simplify it:
In order to solve this, we want the polynomial set equal to 0. To do this, subtract 105 from both sides:
The Rational Roots Theorem says that if p/q is a root of the polynomial, then p is a factor of the constant term and q is a factor of the leading coefficient. The constant term is -105. Drawing a factor tree, we find that the factors of this number are 1, -1, 3, -3, 5, -5, 7, -5, 15, -15, 21, -21. The leading coefficient is 1; its only factor is 1. This means p/q must be a whole number, and can be any of the factors of 105.
Using synthetic division, we try 1 in the box:
<u>1 |</u> 1 -18 80 -105
_______<u> 1</u>__<u> -17</u>___<u>63</u>__
1 -17 63 -42
Since there is a remainder, this is not a root. Trying -1,
<u>-1 |</u> 1 -18 80 -105
_______<u>-1</u>__<u> 19 </u>_<u>-99</u>_
1 -19 99 -204
This is not a root; in fact, it shows us that none of the negatives will be a factor, as the absolute values increase as we complete the synthetic division.
Trying 3,
3 | 1 -18 80 -105
________<u>3</u>__<u>-45</u>___<u>105</u>_
1 -15 35 0
Since there is no remainder, 3 is a root, and is the answer we are looking for.