Question 1126946
Find a polynomial of the specified degree that satisfies the given conditions. 
Degree 5; zeros -3, -1, 3 and 
square of 3
; integer coefficients and constant term 54
<pre>
The zeros are:
 
{{{matrix(2,5,
x=-3,   x=-1,  x=3,    x=sqrt(3),       x=-sqrt(3),
x+3=0,  x+1=0, x-3=0,  x-sqrt(3)=0,    x+sqrt(3)=0)}}}

Multiply left and right sides:

{{{(x+3)(x+1)(x-3)(x-sqrt(3))(x+sqrt(3)) = 0}}}

Multiply both sides by a constant A:

{{{A(x+3)(x+1)(x-3)(x-sqrt(3))(x+sqrt(3)) = 0}}}

FOIL out the last two parentheses on the left:

{{{A(x+3)(x+1)(x-3)(x^2-3) = 0}}}

FOIL out the last two parentheses on the left again:

{{{A(x+3)(x+1)(x^3-3x-3x^2+9) = 0}}}

Get the last parentheses in descending order:

{{{A(x+3)(x+1)(x^3-3x^2-3x+9) = 0}}}

Multiply the last two factors in parentheses on the left:

{{{A(x+3)(x^4-2x^3-6x^2+6x+9)=0}}}

{{{A(x^5 + x^4 - 12x^3 - 12x^2 + 27x + 27)=0}}}

Distribute the A:

{{{Ax^5 + Ax^4 - 12Ax^3 - 12Ax^2 + 27Ax + 27A)=0}}}

Since the constant term 27A must equal 54, we set
them equal:

                    27A = 54
                      A = 2

So {{{Ax^5 + Ax^4 - 12Ax^3 - 12Ax^2 + 27Ax + 27A)=0}}}

becomes

{{{2x^5 + 2x^4 - 12(2)x^3 - 12(2)x^2 + 27(2)x + 27(2))=0}}}

 or

{{{2x^5 + 2x^4 - 24x^3 - 24x^2 + 54x + 54 = 0}}}

So the polynomial is the left side without the "= 0".

{{{2x^5 + 2x^4 - 24x^3 - 24x^2 + 54x + 54}}}

Edwin</pre>