Solving a Quadratic Equation: Finding All Linear Factors

AI Thread Summary
To find all linear factors of the polynomial P(x)=2x^3-3x^2-17x+30, the user initially performed synthetic division with x-2, expecting a remainder of zero since it is a factor. However, they encountered a non-zero remainder, prompting confusion. Other users pointed out arithmetic errors in the calculations, specifically in the addition of coefficients. They advised applying the quadratic formula to the resulting quadratic after correcting the mistakes. The discussion highlights the importance of accurate arithmetic in polynomial factorization.
DumbKid88
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Ok this should be pretty simple math for you guys (Sophomore in HS)

Given that x-2 is a factor of P(x)=2x^3-3x^2-17x+30, find all other linear factors.

I started out doing synthetic division, and got 2x^2-x-19 with remainder -8. Shouldn't the remainder be zero since x-2 is a factor?

Please Help!
 
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Too tempted to make a dumb joke, but I'll wait. :biggrin:

PS: Dk88, welcome to PF! Are you having trouble posting? Have you read the forum guidelines?
 
Sorry i accidentally posted it when I was just trying to preview it.
 
Check your arithmetic -17 + 2 \ne -19

Then when you get a quadratic apply the quadratic formula.
 
SnipedYou said:
Check your arithmetic -17 + 2 \ne -19

Then when you get a quadratic apply the quadratic formula.

And that, my friend, is why my name is DumbKid. Ya -3 +4 surely is not -1. Thanx for the help.
 

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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