Factor Theorem: use to factor x^3 + 3x^2 - 18x -40 or to prove...
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Steven G
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Factor Theorem? I don't think that I heard of that one before. Possibly it was made up by your teacher. Can you please tell us what the exact theorem is please. Thanks
Maybe check this site out? http://www.purplemath.com/modules/factrthm.htm
I looked it up to find the exact definition and it says "The factor theorem is a theorem linking factors and zeros of a polynomial. It is commonly applied to factorizing and finding the roots of polynomial equations." Hope this helps!
Alright, so essentially the factor theorem just says that if x = a is a root of the polynomial, then (x - a) is a factor. But that doesn't really help you solve the problem. There are many ways one can find some or all of the roots of a cubic equation, including factoring by grouping, the Rational Root Test, "depressing" the cubic, or even the dreaded cubic formula (although you most likely won't want to use the cubic formula; it really is, truly, awful).
However, I don't see how any of these methods could be classed as "using the factor theorem." What, specifically, did your class examples do to solve these problems? Do you just "brute force" it and try x=0, x=1, x=-1, x=2, etc. until you "strike gold?" In any case, what happens when you try to follow your class's methods? How far do you get? Where do you get stuck? The more specific you can be, the better we can help you. Even if you know a particular bit of your work is wrong, please include it anyway, just so we can see what you've tried - you might be almost there, just needing a small nudge in the right direction.
However, I don't see how any of these methods could be classed as "using the factor theorem." What, specifically, did your class examples do to solve these problems? Do you just "brute force" it and try x=0, x=1, x=-1, x=2, etc. until you "strike gold?" In any case, what happens when you try to follow your class's methods? How far do you get? Where do you get stuck? The more specific you can be, the better we can help you. Even if you know a particular bit of your work is wrong, please include it anyway, just so we can see what you've tried - you might be almost there, just needing a small nudge in the right direction.
Steven G
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The real question is does it help you? That is the important part. What did you learn from that website?
HallsofIvy
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What I would call the "factor theorem", for polynomials, is:
"Polynomial P(x) has x- a as factor if and only if P(a)= 0".
Here \(\displaystyle P(x)= x^3 + 3x^2 - 18x -40\). Further, the "rational root theorem" tells us that any rational number making P equal to 0 must be an integer that evenly divides 40. Those are 1, -1, 2, -2, 4, -4, 5, -5, 8, -8, 10, -10, 20, -20, 40, and -40.
It is easy to see, by evaluating, that P(-2)= -8+ 12+ 36- 40= 48- 48= 0 so x-(-2)= x+ 2 is a factor.
Dividing P(x) by x+ 2 gives P(x= (x+ 2)(x^2+ x- 20)= (x+ 2)(x- 4)(x+ 5).
"Polynomial P(x) has x- a as factor if and only if P(a)= 0".
Here \(\displaystyle P(x)= x^3 + 3x^2 - 18x -40\). Further, the "rational root theorem" tells us that any rational number making P equal to 0 must be an integer that evenly divides 40. Those are 1, -1, 2, -2, 4, -4, 5, -5, 8, -8, 10, -10, 20, -20, 40, and -40.
It is easy to see, by evaluating, that P(-2)= -8+ 12+ 36- 40= 48- 48= 0 so x-(-2)= x+ 2 is a factor.
Dividing P(x) by x+ 2 gives P(x= (x+ 2)(x^2+ x- 20)= (x+ 2)(x- 4)(x+ 5).