Find a quadratic polynomial divisible by (2x-1) and (x+3) and which leaves remainder 12 on division by (x-1) - Maths - Polynomials

Question

Find a quadratic polynomial divisible by (2x-1) and (x+3) and which
leaves remainder 12 on division by (x-1)

Vijay Kumar Gupta · Accepted Answer

Let the required polynomial be     fx=ax2+bx+cThe division algorithm states that if a function fx is divided by gx, thenthere exist qx and rx such that,   fx=gx qx+rxHere  qx is the quotient term and rx is the remainder term
It is given that the function is divisible by 2x-1 and x+3
So the remainder in this case is 0The corresponding zeroes are    2x-1=0⇒x=12and    x+3=0⇒x=-3Also, the function when divided by x-1  gives 12 as remainderBy remainder theorem if fx is divided by x-a, then remainder is faThis further implies that,     f12=0, f-3=0  and f1=12The three equations thus obtained are                          f12=0     a122+b12+c =0    a14+b12+c =0                    a+2b+4c=0         
 iand                        f-3=0     a-32+b-3+c =0                   9a-3b+c=0         
 iiand                      f1=12     a12+b1+c =12                      a+b+c=12         
 iiiMultiply ii by 4 and then subtract i from the resultant to get     36a-12b+4c=0          a + 2b +4c=0       -   -       -                 35a-14b=0            
 ivSubtract iii from ii to get        9a- 3b+c=0          a + b +c=12       -   -    -                  8a-4b=-12    ⇒   2a-b=-3           
 vMultiply v  by 14 and subtract it from iv   35a-14b=0    28a-14b=-42   -      +         +         7a=42       a=6From v, the value of b can be calculated as      b=2a+3=26+3=12+3=15Put the values of a and b in iii  to get, 6+15+c=12 21+c=12 c=12-21 c=-9Therefore the required quadratic polynimial is         fx=6x2+15x-9

Find a quadratic polynomial divisible by (2x-1) and (x+3) and which leaves remainder 12 on division by (x-1).