Introduction
Working out questions involving polynomials requires to apply the laws of operations i.e. make sure brackets are worked out first, followed by of, followed by division, followed by multiplication, followed by addition, and then multiplication. Therefore, in the expression P (1+ r/2)2, p cannot be multiplied directly to the terms in brackets prior to squaring them. Likewise (-9x3 +3x2 + 15x)/ (-3x), is simplified by factoring the numerator and then dividing it with the denominator.
The polynomial which is used to solve this problem of P invested for one year in a situation where interest compounded semi annually is P(1 + r/2)2. This polynomial can be expressed without using parenthesis using the following steps
Step 1: P*(1 + r/2)2 showing P multiplied by the squared quantity
Step 3: p (1+r/2+ r/2+ r2/4) the expression after FOIL was carried out
Step4: p (1 + r + r2/4) grouping like terms together and combing r/2 +r/2 = r
Step 5: p + pr +pr2/4 multiplying p with all terms in brackets i.e. across the trinomial to obtain a trinomial of variable r which is not in descending order but ascending order (Prasolov, (2010).
The above polynomial can be used to solve a P= 200 invested at interest rate of 11% for 1 years compounded semiannually as follows
Step 1: write 11% in decimal form i.e. 0.11
Step 2: substitute all the terms in the polynomial p + pr +pr2/4 as 200 + 200(0.11) +200(0.11)2/4
In the above step 0.112= 0.0121
Step 3: square and multiply where necessary 200+ 22+0.605= 222.605
Therefore, if $ 200 is compounded semi annually for one year the answer is $ 222.605.
Using the same steps a principle of 5670 invested for one year compounded semi annually can be directly obtained by substituting the terms as follows
Trinomial to be used p + pr +pr2/4
Substituting terms 5670+ 5670 (0.035) + 5670 (0.035)2/4
In the above step 0.352 = 0.001225; when divided by 4 (0.001225/4) = 1.7364 while 5670 (0.035) = 5670 *0.035= 198.45
Addition 5670 + 198.45+ 1.7364 = 5870.2
Therefore, $ 5670 compounded semi annually for one year at 3.5% interest rate results to $ 5870.2.
The above calculations are done by a person who is depositing money in a financial institution. The investor applies the above calculations to know the amount of money which will have been saved after a given duration of time. Likewise, when obtaining a loan a person uses the above formulae to find out the amount of money payable after a given duration of time
Problem 70 on page 311
(-9x3 +3x2 + 15x)/ (-3x). This is an expression showing (-9x3 + 3x2 + 15x) is divided by (-3x). The division can be carried out using the following steps. Therefore the numerator is (-9x3 +3x2 + 15x) and the denominator which is also the divisor is -3x (Prasolov, 2010).
Factoring out -3x does not change the value of the numerator because this is just taking a common factor -3x outside the brackets.
Step two: write the quotient with the factored denominator -3x( 9x2 –x + 5)-3x I.e. -3x (9x2 – x - 5)
Divided by -3x.
Step three: divide by cancelling -3x in the numerator by -3x in the denominator -3x( 9x2 –x + 5)-3x to get 9x2-x +5 which is a trinomial of variable x in descending order.
Trinomials of this kind are applicable in calculating total revenue where x can be taken as price per item. The polynomial can be factored to obtain two values of x which are known as roots of the polynomial. Such roots indicate the least number of items which can be sold to make zero profit and the highest number of items which can be sold to make highest profit.
Conclusion
The expression P(1 + r/2)2 can be expanded to obtain a trinomial p + pr +pr2/4 which is in descending order. The expansion is carried out without changing the value of any variable. This trinomial is used in calculating amount of money accumulated when a fixed amount (Principle P) is saved at a given rate of interest rate r. In addition, it can be used to calculate the amount of money payable after a given period if a loan is obtained from a financial institution at a given interest rate. In case of a loan P is represented by the loan obtained. In evaluating (-9x3 +3x2 + 15x)/ (-3x), the division is carried out in away to ensure the value of terms is not altered. This involves careful process of factoring terms in the numerator.
References
Prasolov, V. V. (2010). Polynomials ([1st softcover print.]. ed.). Berlin: Springer.
