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-60x^5+77x^4+6119x^3-4044x^2-99952x-24640

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3 Answers

We want to factor the quintic polynomial p(x) = - 60x^5 + 77x^4 + 6119x^3 - 4044x^2 - 99952x - 24640.  

We will make use of the following fact.  If p(x) is a polynomial, and a is a real number such that p(a) = 0, then (x-a) is a factor of p(x).  

With this in mind, we set a to be a few small integers, and compute p(a), i.e. p(0), p(±1), p(±2), p(±3), p(±4), ... , p(±10).   

We find that p(5) = p(-4) = 0.  So (x-5) and (x-(-4)) = (x+4) must be factors of p(x), i.e. p(x) = (x-5)(x+4) q(x), where q(x) is some polynomial.  

Compute q(x) = p(x) / [(x-5)(x+4)] = - 60 x^3 + 17 x^2 + 4936 x + 1232.  

Now set a to be a few simple rational numbers, and compute q(a), i.e. q(±1/2), q(±1/3), q(±2/3), q(±3/2), q(±1/4).  

We find that q(-1/4) = 0.  So (4x + 1) is a factor of q(x), i.e. q(x) = (4x + 1)r(x), where r(x) is some polynomial.  

Compute r(x) = q(x) / (4x + 1) = - 15x^2 + 8 x + 1232, which factors to - (5x + 44)(3x - 28).  

So p(x) = (x-5)(x+4) q(x) = (x-5)(x+4)(4x + 1)r(x) = - (x-5)(x+4)(4x + 1)(5x + 44)(3x - 28).  

Note: The reason this approach worked is that of the 5 possible complex roots of the quintic p(x), at least 3 were rational.  

Simple. Use the rational-root test. It says that for polynomials with integer coefficients, if the leading coefficient is L and the constant coefficient is C, and Q is a rational root (if one exists) then Q=K/N where K is a factor of C and N is a factor of L. In the case of this quintic polynomial, we have L=-60 and C=-24640. Thus the candidates are "+/- (factor of -24640) / (factor of -60)". Unfortunately, there are a lot of factors for -24640, and -60 so there are lot's of candidates.

Comments

Also if Q=K/N in lowest terms is a rational root, then Nx-K is a factor of the polynomial. For the given quintic, all roots are rational, so you get 5 linear factors.

Comment

Having a problem submitting the fully calculated equation because the system is wanting to recognize the numbers as phone numbers instead of just answers to the problem; and won’t let me submit the answer like that. So… I am going to break this down a little differently and maybe WyzAnt will let me post the whole worked out numbers later. Until then this is the best I can do Split into two separate equations, solve, and put back together at the end (and use quadratic) -60x^5 + 77x^4 + 6,119x^3 - 4,044x^2 - 99,952x - 24,640 Becomes -60x^5 + 77x^4 + 6,119x^3 OR -Ax^5 + Bx^4 + Cx^3 And -4,044x^2 - 99,952x - 24,640 OR -Dx^2 - Ex - F #1: -Ax^5 + Bx^4 + Cx^3 -x^3(-Ax^2 - Bx - C ) pull the –x^3 out and save to put back on later {B +- v[(B^2) -(4AC)]} / 2A Simplify down 10.76 AND -9.477 -x^3(x - 10.76)(x + 9.477) put into ( ) and add the –x^3 back on #2: -Dx^2 - Ex - F -(Dx^2 + Ex + F) pull a (-) out and save to put back on later {-D +- v[(D^2) -(4AC)]} / 2D Simplify down -0.249 AND -24.467 -(x + 0.249)(x + 24.467) put into ( ) and add the (–) back on Put the two pieces back together -x^3(x - 10.76)(x + 9.477) -(x + 0.249)(x + 24.467)