Question 1176598
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Call the expression {{{p(x) = 2x^3+ax^2+bx-2}}}.<br>
(x-2) being a factor of p(x) means that the remainder when p(x) is divided by (x-2) the remainder is 0; that means<br>
{{{p(2) = 2(2^3)+a(2^2)+b(2)-2=0}}}
{{{16+4a+2b-2=0}}}
{{{4a+2b=-14}}}
{{{2a+b = -7}}}  [1]<br>
Similarly, if p(x) divided by (x-3) leaves remainder -50, that means<br>
{{{p(3) = 2(3^3)+a(3^2)+b(3)-2 = -50}}}<br>
{{{54+9a+3b-2 = -50}}}
{{{9a+3b = -102}}}
{{{3a+b = -34}}}  [2]<br>
Solve equations [1] and [2] to find a=-27 and b=47.<br>
The expression is 2x^3-27x^2+47x-2.<br>
We know one of the roots is 2; synthetic division leads us to<br>
{{{2x^3-27x^2+47x-2 = (x-2)(2x^2-23x+1)}}}<br>
The quadratic factor has irrational roots; you can find them exactly using the quadratic formula; or you can find them approximately using a graphing calculator or similar tool.<br>
I'm guessing the a), b), and c) in your post are supposed to be the answer choices; but none of them is correct....<br>