Factor Theorem Question

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1. Mar 29, 2015

Rithikha

1. The problem statement, all variables and given/known data
If the polynomial P(x) = x^2+ax+1 is a factor of T(x)=2x^3-16x+b, find a, b

2. Relevant equations

3. The attempt at a solution
Let (px+q) be a factor of P(x),
p can possibly be 1 and so can q, according to factor theorem,
Hence, factors (x+1) or (x-1)
P(1) = 0, substituting I got -2 as a and P(-1) =0 , I got 2 as a
If either (x+1) or (x-1) is a factor of P(x), it has to be a factor of T(x),
T(1) = 0, I got 14 as b in both cases.
But the correct answer is a=3,-3 and b=-6,6 respectively.

2. Mar 29, 2015

SammyS

Staff Emeritus
Hello Rithikha. Welcome to PF !

You want $\ p\, x +q \$ to be a factor of $\ T(x)\$ rather than $\ P(x)\$.

3. Mar 29, 2015

Rithikha

But the constant term is b in T(x), we can get q from it. Or can we?

4. Mar 29, 2015

HallsofIvy

Staff Emeritus
Your error is in assuming that P(x) has a linear factor with rational coefficients (which you do when you use the rational root theorem).

Instead let y= P(x) and rewrite T(x) in terms of y.

5. Mar 29, 2015

Ray Vickson

You want $2x^3-16x + b = (d x + c)(x^2 + ax + 1)$. It is easy to see that we must have $d = 2$, so we need $2x^3 - 16 x + b = (2x + c)(x^2 + ax +1)$.