6. The rabbits will never die.
The question was how many male/female rabbit pairs will be there after a year or 12 months?
When the experiment begun, there is a single pair of rabbits.
After duration of one month, the two rabbits have mated though they have not given birth. As a result; there is still only a single pair of rabbits.
After duration of two months, the initial pair of rabbits will give birth to another pair. There will be two pairs.
After duration of three months, the initial pair will give birth again, the second pair mate, but do not give birth. This makes three pair.
When four months will elapse, the original pair gives birth, and the pair born in the second month gives birth. The pair that is born in month in the third month will mate, but will not give birth. This will make two new pairs, thereby making a total of five pair.
After duration of five months, each pair that was alive two months earlier will give birth. This will make three new pair, totaling to eight (Anderson, Frazier, and Popendorf, 1999)
Factor Theorem:
Value a is a root of polynomial p (x) when and only when (***) is a factor of p (x).
Evidence
1. (=)) Assuming that a is one of the roots of polynomial p (x). This implies that p (a) = 0.
Using the remainder theorem, we can conclude that remainder after being divided by (***) has to be zero. Therefore,
P (x) = (***) ? q (x) + 0
P (x) = (***) ? q (x)
Hence, (***) is a factor of p (x).
To establish the factors of a polynomial one can speedily substitute in values of x to find out which will provide you with a value of zero.
Example 1: Given f (x) = x3 + x2 — 4x — 4. Use the factor theorem to find a factor f (x) = x3 + x2 — 4x — 4
f (1) = (1)3 + (1)2 — ( ) — 4
(x — 1) is not a factor f (2) = (2)3 + (2)2 — ( ) — 4
= 0 ?(x — 2) is a factor
To move away from the value x to the factor merely place it into a bracket and then change the sign.
Rational Root (zero) Theorem
The Rational Zero Theorem provides a list of probable rational zeros of the polynomial function. The theorem provides all potential rational roots of the polynomial equation. Not all numbers in the list shall be a zero belonging to the function, however all rational zeros belonging to the polynomial function shall come out somewhere in the list.
The rational root theorem is also a test that is capable of being used to get the probable number of rational solutions or sometimes roots of the polynomial equation having coefficients which are integers.
Degree
The degree of a monomial equals to the sum of the exponents of individual variables that appears in the monomial. For example, the degree of x2yz3 is 2 + 1 + 3(Beckmann, 1976)
A polynomial is a monomial. It can also be said to be the algebraic sum or sometimes the difference of monomials.
A polynomial degree is the greatest of the degrees of its terms after the combination of like terms. The leading coefficient is described as the coefficient of the term with the greatest. The polynomials which are having one, two or even three terms are referred to as monomials, binomials and trinomials in that order (Buchanan, 2010)
The degree of a monomial can simply be defined as the exponent or power that the monomial is raised to. If there exists three or more monomials that are being added or subtracted so as to make a polynomial, and each of them has a degree and the monomial having the highest degree are representing the whole degree of the polynomial.
The fundamental theory of algebra
It is one of the most essential results in mathematics.
The Fundamental Theorem of Algebra is practically basic spontaneously. It states that provided with any polynomial that is not constant in the field C, we can always get a root of the polynomial that is provided. In spite of the fact that it is scarcely applied directly in mathematical proofs, it is always an essential feature to the other theorems. For example, in showing evidence that all angles cannot be trisected by use of a straight-edge and compass only, the Fundamental Theorem of Algebra is applied if the roots of any polynomials are got. The theorem is what makes the other big theorems in math to work, like the Hilbert Nullstelensatz Theorem. Additionally, also the Axiom of Choice applies the Fundamental Theorem of Algebra if proven by Zorns Lemma, which was introduced in the year1935, and which is similar to the Axiom of Choice (Fraleigh, 2003)
The Theorem was proved first by Carl Frieddrich Gauss (1777-1855).The Fundamental Theorem of Algebra tells us that when we have completely factored a polynomial: On one side, the polynomial has completely been factored if only all its factors are linear or are irreducible quadratic. On the contrary, every time a polynomial has been factored into linear or irreducible quadratics only, then it has completely been factored because both linear factors and irreducible quadratics are not capable being factored any more over real numbers. The theorem is however not constructive since it does not tell us the way to factor completely a polynomial.
References
Buchanan, R. (2010). Addition and subtraction with polynomials, http://banach.millersville.edu/~bob/math101/AddSubPoly/main.pdf, assessed on February, 24, 2010
Anderson, M; Frazier, J and Popendorf, K. (1999). The Rabbit Problem,
http://library.thinkquest.org/27890/theSeries2.htm. Assessed on February 24, 2011
Beckmann, P. (1976). A History of Pi, St. Martins Griffin.
Brousseau (1969). “Fibonacci Statistics in Conifers.” Fibonacci Quarterly (7): 525 — 532.
Fraleigh, J. (2003). A First Course in Abstract Algebra Seventh Edition, Addison
Wesley, Boston.
Fuzzy, N.(2010). The fibonacci patterns in nature? Retrieved from http://www.mymathforum.com/viewtopic.php?f=43&t=18021 .
Assessed on February,24,2011
Goonatilake, S (1998). Toward a Global Science.Indiana University Press. p. 126.
. http://books.google.com/?id=SI5ip95BbgEC&pg=PA126&dq=Virahanka+Fibonacci
Grist, S. (n.d.) fibonacci numbers in nature and the golden ratio. Retrieve from: http://www.world-mysteries.com/sci_17.htm#Nature Assessed on February, 24, 2011
Jones, Judy; William Wilson (2006). “Science.” An Incomplete Education. Ballantine Books.
p. 544. a.
Knuth. (2006).The Art of Computer Programming: Generating All Trees- History of Combinatorial Generation; Volume 4. Addison-Wesley. p. 50.
http://books.google.com/?id=56LNfE2QGtYC&pg=PA50&dq=rhythms.
Laurence, S.E. (2002). Fibonaccis Liber Abaci. Springer-Verlag. Chapter II.12, pp
404 –.