- #1
Psybroh
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Hi, I've been trying a couple of proofs that my calc teacher gave me, but I'm not sure if I have the right approach or not.
1) Prove that the degree of the depressed polynomial is exactly one less than the degree of the original polynomial.
- For this proof, all I can come up is the face that since a "x" has been removed from P(x), the depressed polynomial Q(x) has 1 less "x" in each of its terms, and therefore has one less degree. is this correct? I cannot seem to express this in terms of variables and numbers.
2) Factor P(x) = x^3 + x^2 - 16x + 20 into the product of a constant and 3 linear factors.
- Can I factor out a 4 and use that as a constant?
3) Show that x=a is a root of x^3 - ax^2 + ax - mx^2 - a^2 + amx = 0, without using synthetic division.
- Do I just plug in x=a into the polynomial and show that the entire thing does come out to zero, or...?
4) Given that x=a is a root of x^3 - ax^2 - ax - mx - mx^2 + a^2 + am + amx = 0, use synthetic division to factor that eight-term polynomial into the product of two factors: one in linear x, one quadratic in x.
- I have NO idea how to even start this... any help would be great! lol
Well, these are it. I have some ideas to solve them, but I'm not sure if they can really be considered proofs. Any help and suggestions would be great! Thanks! :)
1) Prove that the degree of the depressed polynomial is exactly one less than the degree of the original polynomial.
- For this proof, all I can come up is the face that since a "x" has been removed from P(x), the depressed polynomial Q(x) has 1 less "x" in each of its terms, and therefore has one less degree. is this correct? I cannot seem to express this in terms of variables and numbers.
2) Factor P(x) = x^3 + x^2 - 16x + 20 into the product of a constant and 3 linear factors.
- Can I factor out a 4 and use that as a constant?
3) Show that x=a is a root of x^3 - ax^2 + ax - mx^2 - a^2 + amx = 0, without using synthetic division.
- Do I just plug in x=a into the polynomial and show that the entire thing does come out to zero, or...?
4) Given that x=a is a root of x^3 - ax^2 - ax - mx - mx^2 + a^2 + am + amx = 0, use synthetic division to factor that eight-term polynomial into the product of two factors: one in linear x, one quadratic in x.
- I have NO idea how to even start this... any help would be great! lol
Well, these are it. I have some ideas to solve them, but I'm not sure if they can really be considered proofs. Any help and suggestions would be great! Thanks! :)