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Questions of exercise 4.3

Find the roots of the following equations 

            

(a)  X -   1    = 3     (b)    1     -       1      =   11 
               X                    X+4         X-7           30

Question No.4...The sum of the reciprocals of Rehman's age(in years) three years ago and five years from now will be 1/3. Find his present age.

Solution: Let Rehman's present age= x
   
                  3 years ago
             Rehman's age=x-3

                5 years After
              Rehman's age= x+5



By the question

   1     +          1    =   1
x-3             x+5        3
Cross multiplication
x+5 + x -3     =    1 
(x-3)(x+5)           3

   2x + 2               =    1 
x² + 5x-3x -15       3

   2x + 2        =      1 
x² +2x -15         3

x² +2x -15   = 6x + 6

x² +2x -15 - 6x - 6 =0

x² - 4x - 21 = 0
Now splitting the middle term

x² -7x +3x -21 = 0

x(x-7) +3(x-7) = 0

(x-7) (x+3) = 0

Either x-7 =0     or    x+3=0
         =>   x=7     or     x = -3  (Rejected)
                           because age can't be negative

Hence Rehman's present age = 7years

Types of polynomials

Polynomials can be divided on the basis of degree and on the basis of number of terms... 

 On the bases of degree.. On the basis of degree polynomials can be divided into linear,quadratic,cubic,biquadratic, quintic,sextic....e.t.c 

 LINEAR POLYNOMIAL: A polynomial is said to be linear polynomial if it's degree = 1 e.g (x+1) 

 Quadratic Polynomial: A polynomial is said to be quadratic polynomial if it's degree =2 e.g x^2 +1....read as (x square plus 1) 

 CUBIC POLYNOMIAL: A polynomial is said to be cubic polynomial if it's degree =3.. e.g x^3 +1..... read as (x cube plus 1)

 BIQUADRATIC POLYNOMIAL: A polynomial is said to be biquadratic polynomial or (Quartic polynomial) if it's degree = 4 e.g x^4 +5...read as x raised to power 4 plus1

 On the basis of number of terms On the basis of number of terms polynomials can be divided into MONOMIALS BINOMIALS TRINOMIALS e.t.c

 MONOMIAL: A polynomial is said to be monomial if it contains only one term...MONO means 1 e.g...2x 5y,7z 

 BINOMIAL:A polynomial is said to be binomial if it contains only two terms...BI means Two.. e.g 2x+3y, 7z+6x etc 

 TRINOMIAL: A polynomial is said to be trinomial if it contains only three terms...Tri means 3... e.g 7x+3y+2z

Method of completing squares

Step 1 Divide all terms by a (the coefficient of x square).

Step 2 Move the number term (c/a) to the right side of the equation.

Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation. We now have something that looks like (x + p)2 = q, which can be solved rather easily:

Step 4 Take the square root on both sides of the equation. Step 5 Subtract the number that remains on the left side of the equation to find x.