If a and c are odd ' numbers

poznateqojh

poznateqojh

Answered question

2022-04-09

If a and c are odd ' numbers and ax2+bx+c=0 has rational roots, where bI, then prove that one root of the equation will be independent of a and b.

Answer & Explanation

zvonkurm7h

zvonkurm7h

Beginner2022-04-10Added 8 answers

Step 1
Since a and c are odd ' numbers, the Rational Zeroes Theorem proposes the candidates
 x = ±ca , ±c1 , ±1a , ±11  .
The factorization of the quadratic polynomial is  ax2 + bx + c = a·(xr)·(xs)  , so the four factorizations that will produce the correct leading and constant terms are
a·(xca)·(x1)  ,  a·(x+ca)·(x+1)  ,  a·(x1a)·(xc)  ,
a·(x+1a)·(x+c)  .
Step 2
The "middle coefficient"  b = -a·r+s in these products of factors is one of
a·(±ca ± 1)  =  ± (c+a)   or   a·(±1a ± c)  =  ± (1+ac)  ,
any of which is an integer.
Hence, one of the rational roots of the quadratic equation is ±1 or ±c, which are independent of a and b.

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