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
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
so do u succeed??
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)]}
ok, i think i get it now. thank you!
i got it. thanks a lot*
Sure, I'd be happy to help you factor the expression x^10 + x^5 + 1 step by step.
Step 1: Recognize the Pattern
Notice that the given expression resembles a geometric series with a common ratio of x^5. In a geometric series, the sum of n terms can be written as (1 - r^n) / (1 - r), where r is the common ratio. We can use this pattern to factor the expression.
Step 2: Substitute Variables
Let's substitute a new variable, let's say y, to represent x^5. So, y = x^5.
Step 3: Rewrite the Expression
Using the substitution, we can rewrite the expression as:
x^10 + x^5 + 1 = y^2 + y + 1
Step 4: Factor the Expression
Now, we need to factor the quadratic expression y^2 + y + 1. This can be done by using the quadratic formula or by completing the square. However, in this case, this quadratic does not factor nicely into linear binomials with integer coefficients.
So, the expression x^10 + x^5 + 1 cannot be factored into simpler terms using elementary algebraic methods without leaving the expression as a quadratic.
However, if you are allowed to use complex numbers, you can factor it further using the concept of roots of unity and express it as:
x^10 + x^5 + 1 = (x^10 - x^5 + 1) / (x^5 - 1)
This form provides more insights but might not be considered a "simplified" form.
Make sure to mention that the expression cannot be factored further using elementary algebraic methods in your homework explanation.