Help! Factor Trinomial with X^3: Sample Problem Included

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To factor the trinomial x^3 - 2x^2 + 1, it is suggested to start by using the rational root theorem to identify potential factors. Another method mentioned is factoring by grouping, which involves rearranging the polynomial into two groups: x^3 - x^2 and -x^2 + 1. This allows for further simplification and identification of common factors. The initial attempt to factor out x was incorrect, as it did not yield the original polynomial. Understanding these techniques can help in successfully factoring similar polynomials.
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Homework Statement



I need help trying to factor a trinomial. It has been a while, and I can't remember how to factor a trinomial with x^3. Please help.

Sample problem...

X^3 - 2X^2 + 1

Thanks



Homework Equations





The Attempt at a Solution



Not sure where to start, I would think I would factor out an X

Maybe: X^2 (1X - 2 + 1)

?
 
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jacksonbobby5 said:

Homework Statement



I need help trying to factor a trinomial. It has been a while, and I can't remember how to factor a trinomial with x^3. Please help.

Sample problem...

X^3 - 2X^2 + 1

Thanks



Homework Equations







The Attempt at a Solution



Not sure where to start, I would think I would factor out an X

Maybe: X^2 (1X - 2 + 1)

?

well I would say x^2(x-2) +1 insteed. But maybe its just me...
 
jacksonbobby5 said:

Homework Statement



I need help trying to factor a trinomial. It has been a while, and I can't remember how to factor a trinomial with x^3. Please help.

Sample problem...

X^3 - 2X^2 + 1

Thanks



Homework Equations





The Attempt at a Solution



Not sure where to start, I would think I would factor out an X

Maybe: X^2 (1X - 2 + 1)
This is incorrect. If you simplify the expression in parentheses, you get x - 1. If you then multiply x^2 and x -1, you get x^3 - x^2, which is different from what you started with.

If you are being asked to factor polynomials such as this one, it's possible that you have learned about the rational root theorem. It gives you a way to find the potential factors of the polynomial, based on the coefficients of the highest and lowest degree terms in the polynomial.

If you haven't learned this theorem, your polynomial can still be factored using another technique called factoring by grouping.

x^3 - 2x^2 + 1 = x^3 - x^2 - x^2 + 1

Group together the first two terms on one group, and the last two terms in another group. Can you continue from here?
 

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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