<p>I am trying to solve this problem right now and I have NO IDEA how to solve it rationally. You are not allowed to use a calculator, which is killing me. Can any math-minded people out there please explain how you solve this infernal problem?</p>
<p>The polynomial p(x) = 4x^3 − ax^2 − 41x + b has zeros at x = −1/2, 3, and −7/2. What is the product of a and b?
(a) −55 (b) 50 (c) 0 (d) 60 (e) 84</p>
<p>Ahhhh it is driving me crazy! Please help!</p>
<p>if roots are r1, r2, r3; p(x) = k(x-r1)(x-r2)(x-r3)</p>
<p>Plug in your roots and you will see setting k equal to 4 cleans up the 1/2 and 7/2 to yield</p>
<p>p(x) = (2x + 1)(x-3)(2x+7)</p>
<p>We could FOIL away and gather terms to find a and b, but it easier to note that b = -21 and that -a = 14 + 2 -12 = -4; </p>
<p>so ab = 84</p>
<p>ahhh thank you thank you</p>
<p>Alternatively you could plug in two of the roots into the equation and set P(x) to 0, this will get you a system of equations with two variables and two equations. Then solve for a and b.</p>