Question 125808

A polynomial of degree {{{3}}} :  {{{ax^3 + b x^2 + cx + d=0}}}

The coefficient of x^2 ({{{b}}}) is 3 less than the coefficient of x^3 ({{{a}}})

{{{a = b + 3}}}

The coefficient of x ({{{c}}})  is three times the coefficient of x^2 ({{{b}}})

{{{c = 3b }}}
	
The remaining coefficient ({{{d}}})   is 2 more than the coefficient of x^3 ({{{a}}}) .

{{{d = a + 2 }}}
	

The sum of the coefficients is -4.

{{{a+b+c+d=-4 }}}


 
{{{c = 3b }}}……………….(2)
{{{d = a + 2 }}}……………..(3)
{{{a+b+c+d=-4 }}}…………(4)


{{{a = b + 3}}}……………(1)..=>….plug in (3)
{{{d = a + 2 }}}……………..(3)
{{{d = b + 3 + 2 }}}…
{{{d = b + 5 }}}………………….(I)

{{{a+b+c+d=-4 }}}…………(4).substitute {{{a}}} from (1), {{{c}}} from (3), and {{{d}}} from (I)


{{{( b + 3) + b + 3b  + (b + 5 ) = -4 }}}……solve for {{{b}}}

{{{6 b + 8 = -4 }}}……

{{{6 b = -4-8 }}}……

{{{6 b = -12 }}}……

{{{b = -2 }}}……

Now find {{{a}}}

{{{a = b + 3}}}……………(1)

{{{a = -2 + 3}}}
{{{a = 1}}}

Now find {{{c}}}

{{{c = 3b }}}……………….(2)
{{{c = 3(-2) }}}
{{{c = -6 }}}

Now find {{{d}}}

{{{d = b + 5 }}}………………….(I)
{{{d = -2 + 5 }}}……
{{{d = 3 }}}……

So, your coefficients are:

{{{a = 1}}}
{{{b = -2 }}}……
{{{c = -6 }}}
{{{d = 3 }}}……


A polynomial of degree {{{3}}} will be:  {{{x^3 -2x^2 -6x + 3 = 0}}}