Question

Case study 1
Ankur and Ranjan start a new business together. The amount invested by both
partners together is given by the polynomial p(x) = 4 + 12 + 5, which is the 2

product of their individual shares.
i)Find the Coefficient of in the given polynomial and name the polynomial of the 2
amount invested by each partner
ii)Find the shares of Ankur and Ranjan invested individually and also find the total
amount invested by both, if x=1000

Answer 1

i) Coefficients are 4 and 12 of $x^{2}$ and x respectively. Quadratic polynomial.

ii) Individual shares of Ranjan and Ankur are 2001 and 2005.

Total = 4006.

Step-by-step explanation:

i)The coefficients of the given polynomial are 4 of $x^{2}$ and 12 of x. And as the degree of polynomial is 2 the polynomial is a quadratic polynomial.

ii)Given,

$p(x) = 4x^{2} +12x +5$

Also it is mentioned that the given polynomial is the product of shares of Ranjan and Ankur.

By using the method of splitting the middle term in polynomial we can find the individual shares of Ranjan and Ankur.

$4x^{2} +12x+5 = 4x^{2} +2x+10x+5 = 2x(2x+1) + 5(2x+1) = (2x+5)(2x+1)$

As we obtained two different polynomials which are (2x+5) and (2x+1), these are the shares of Ranjan and Ankur.

If the value of x is 1000 ----(given)

by substituting the value of x in the individual shares we obtain that,

$2x+5 = 2(1000)+5 = 2000+5 = 2005$

$2x+1 = 2(1000)+1 = 2000+1 = 2001$

Therefore, the individual shares are 2005 and 2001.

As, we have obtained individual shares we can get the total amount invested by both by adding these two individual shares.

$2005+2001 = 4006$.

Therefore, the total amount invested is 4006.

Answer 2

i) Coefficients are 4 and 12 of x^{2}x

2

and x respectively. Quadratic polynomial.

ii) Individual shares of Ranjan and Ankur are 2001 and 2005.

Total = 4006.

Step-by-step explanation:

i)The coefficients of the given polynomial are 4 of x^{2}x

2

and 12 of x. And as the degree of polynomial is 2 the polynomial is a quadratic polynomial.

ii)Given,

p(x) = 4x^{2} +12x +5p(x)=4x

2

+12x+5

Also it is mentioned that the given polynomial is the product of shares of Ranjan and Ankur.

By using the method of splitting the middle term in polynomial we can find the individual shares of Ranjan and Ankur.

\begin{gathered}4x^{2} +12x+5 = 4x^{2} +2x+10x+5 = 2x(2x+1) + 5(2x+1) = (2x+5)(2x+1)\\\end{gathered}

4x

2

+12x+5=4x

2

+2x+10x+5=2x(2x+1)+5(2x+1)=(2x+5)(2x+1)

As we obtained two different polynomials which are (2x+5) and (2x+1), these are the shares of Ranjan and Ankur.

If the value of x is 1000 ----(given)

by substituting the value of x in the individual shares we obtain that,

2x+5 = 2(1000)+5 = 2000+5 = 20052x+5=2(1000)+5=2000+5=2005

2x+1 = 2(1000)+1 = 2000+1 = 20012x+1=2(1000)+1=2000+1=2001

Therefore, the individual shares are 2005 and 2001.

As, we have obtained individual shares we can get the total amount invested by both by adding these two individual shares.

2005+2001 = 40062005+2001=4006 .

Therefore, the total amount invested is 4006.