Question

50 points. If all the coefficients of a polynomial are positive, then are all the roots negative? Explain(Don't explain if it is just a fact) (Assume that roots are real)

Answer 1

Let the quadratic equation we will deal with is ax²+bx+c=0

here a,b,c are positive.....

So sum of roots = -b/a

Product of roots = c/a

hence sum is negative it is evident that roots are negative.

Let us look at a demostration with an example

x²+6x+9=0


so sum of roots = -6/1=-6

product of roots=9/1

later we find (x+3)²=0

x= -3,-3

hope helped !

Answer 2

★ QUADRATICS RESOLUTION ★

GENERAL QUADRATIC EQUATION →→→

ax² + bx +c =0

Here, coefficients of polynomial are positive

For possible combination about negative roots conclude that a>0, b>0 , where c resides as constant ...

Acquired behaviour of quadratic function is
b² - 4ac =0
so that we can obtain perfect square as roots
in variable x .
it's applicable condition for acquiring perfect squares , with both the roots either positive or negative

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