2. The area of the soccer field can be represented by the binomial 100x^2+4000 where x is measured in metres. a. Factor the polynomial that represents the area to find the length and width of the field . 12 marks] b. FIFA rules say that a standard soccer field is no longer than it is wide. If x=6 does the soccer fit within FIFA rules? [1 mark]

**a. Factoring the Polynomial:**<br /><br />The given area is represented by the polynomial 100*x² + 4000. We can factor this by finding the greatest common factor (GCF) of both terms. The GCF of 100 and 4000 is 100.<br /><br />So, we can factor out 100:<br /><br />100*x² + 4000 = 100(x² + 40)<br /><br />Since x represents a measurement in meters, the factored form 100(x² + 40) represents the area as a product of two dimensions. We can interpret these as length and width. While x² + 40 cannot be factored further using real numbers, we can still express the length and width in terms of x.<br /><br />Possible interpretations of length and width:<br /><br />* **Length = 100 meters, Width = (x² + 40) meters**<br />* **Length = 10(x² + 40) meters, Width = 10 meters**<br />* **Length = 20 meters, Width = 5(x² + 40) meters** and so on. There are other possible combinations. Without additional information, we can't definitively say which factor represents length and which represents width.<br /><br />**b. Checking FIFA Rules:**<br /><br />Given x = 6, let's substitute this value into the expressions for length and width we found in part (a). We'll use the first interpretation as an example:<br /><br />* Length = 100 meters<br />* Width = (x² + 40) meters = (6² + 40) meters = (36 + 40) meters = 76 meters<br /><br />In this case, the length (100 meters) is *longer* than the width (76 meters). This *does not* fit within FIFA rules, which state the field should not be longer than it is wide.<br /><br />Let's check another possible interpretation: Length = 10 meters and Width = 10(x² + 40) meters.<br /><br />* Length = 10 meters<br />* Width = 10(6² + 40) = 10 * 76 = 760 meters<br /><br />In this case, the length (10 meters) is *shorter* than the width (760 meters). This *does* fit within the FIFA rule.<br /><br />Therefore, whether the field fits FIFA regulations depends on which dimension is considered length and which is considered width. The problem doesn't give us enough information to determine this definitively. We can only conclude that *some interpretations* of the dimensions satisfy FIFA rules while others do not.<br />