factoring

posted by .

could anybody help me figure out step by step how to factor this expression:
x^10+x^5+1(please)
p.s. i need to explain this on the hw

  • factoring -

    try to solve this equation with X=x^5
    so u have x^10+x^5+1--->> X²+X+1
    and it's much easier to solve this equation

  • factoring -

    so do u succeed??

  • factoring -

    As mk-tintin explained, you first solve the equation:

    X²+X+1 = 0 --->

    X = -1/2 +- 1/2 sqrt(3) i. We can write this as:

    X = cos(4/3 pi) +- i sin(4/3 pi) (+- sign reversed relative to previous expression)

    This can be written in exponential form as:

    X = exp(+- 4/3 pi i)



    The next step is to compute the five solutions of x^5 = X:


    x = exp(+- 4/15 pi i + 2/5 n pi i)

    for n = 0, 1,...,4 or some other range of five.

    So we have ten solutions in total, let's denote them by w1, w2, ..., w10. Then:

    x^10+x^5+1 = (x-w1)(x-w2)...(x-w10)

    Finally, you want to rewrite the right hand side in terms of real factors only. To do this,
    you exploit the fact that the roots of a polynomial with real coefficients come in pairs
    that are related under complex conjugation. The two factors corresponding to two
    such roots can be multiplied to yield a real quadratic factor. In this case we can use:

    (x - exp(4/15 pi i + 2/5 n pi i))(x - exp(-4/15 pi i - 2/5 n pi i)) =

    x^2 - 2 cos[(4/15 + 2/5 n) pi] x + 1

    So, the factorization can be written as:

    x^10+x^5+1 = (x^2 - 2 cos(4/15 pi) x + 1)(x^2 -2 cos(2/3 pi) x + 1)
    (x^2 - 2 cos(16/15 pi) x + 1)(x^2 - 2 cos(22/15 pi) x + 1)(x^2 - 2 cos(28/15 pi) x + 1)

    Now, we aren't finished yet, because each of the cosine terms can be simplified:

    cos(4/15 pi) = 1/8 [1-sqrt(5)] + 1/4 sqrt{2/3 [5 + sqrt(5)]}

    cos(2/3 pi) = - 1/2

    cos(16/15 pi) = 1/8 [1-sqrt(5)] - 1/4 sqrt{2/3 [5 + sqrt(5)]}

    cos(22/15 pi) = 1/8 [1+sqrt(5)] - 1/4 sqrt{2/3 [5 - sqrt(5)]}

    cos(4/15 pi) = 1/8 [1+sqrt(5)] + 1/4 sqrt{2/3 [5 - sqrt(5)]}

  • factoring -

    ok, i think i get it now. thank you!

  • factoring -

    i got it. thanks a lot*

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. maths

    for this equation x^2 - x = 0 would x = 1 i think this is the answer but i need to write it out step by step in how i worked the answer out, can any one help me please?
  2. Algebra help

    f(x)=x^3-4x^2-x+4 How do I find the x-intercept for this?
  3. Geometry Question plz help

    Could someone please explain how to figure this out step by step. I'll give you one of my problems and I'll try to figure out the rest. Directions: For each situation find the value of x and the measure of the indicated segment. 1. …
  4. Algebra

    4x^4-18x^2=0. I have to find out what x equals by factoring. What I did was Factor 2x^2(2x^2+-9)=0 then 2x^2(2x+3)(x+-3), resulting in -1.5 and three. But these answers are wrong according to my book. Can you show me step by step how …
  5. Math

    Factor 3y^2z-5yz-9y^2+30y-25 I can't seem to figure this one out... Give me a step by step answer please and thank you :)
  6. MATH-ALGEBRA

    What's the slope-intercept equation of line which is parallel to y = 2/5x-3 and passes through point (0,5) A. y = -5/2x - 3 B. y = -5/2x + 5 C. y = 2/5x + 3 D. y = 2/5x + 5 Its D, but how do u figure that out?
  7. chemistry

    This question I really don't get it. I tried tried tried to figure out how to start solve the problem but still couldn't figure out. So can you help me with one question PLEASE?
  8. Quick math help

    In which step below does a mistake first appear in simplifying the expression?
  9. Algebra 1--Step-by-Step

    Can someone show me how to solve these step-by-step?
  10. math

    10a) Simplify the expression and explain each step. 12 + 3(2y – 3) Step 1: apply distributive property Step 2: combine like terms Step 3: simplify b) Factor the expression completely. 18b – 12 What is the GCF of 18 and 12?

More Similar Questions