Roots of a polynomial of degree 4

• Mathman23
So x+ x-1 is a root of Q is equivalent to p(x)= 0. In summary, we are trying to prove that a complex number x satisfies the equation p(x) = 0 if and only if the number s = x + x^-1 is a root of the polynomial Q(s) = s^2 + as + (b-2). This can be shown by observing that Q(x + x^-1) = p(x)/x^2 and noting that for a fraction to equal 0, the numerator must also be 0. Therefore, x + x^-1 is a root of Q if and only if p(x) = 0.
Mathman23
(*)$$p(x) = x^4 + ax^3 + bx^ 2 + ax + 1 = 0$$

where $$a,b \in \mathbb{C}$$

I would like to prove that a complex number x makes (*) true iff

$$s = x + x^{-1}$$ is a root of the $$Q(s) = s^2 + as + (b-2)$$

I see that that $$Q(x + x^{-1}) = \frac{p(x)}{x^2}$$

Then to prove the above do I then show that p(x) and Q((x + ^{-1}) shares roots?

Sincerely Yours
MM23

Yes. Note that Q(s) and p(x) must have coincident roots.

Mathman23 said:
(*)$$p(x) = x^4 + ax^3 + bx^ 2 + ax + 1 = 0$$

where $$a,b \in \mathbb{C}$$

I would like to prove that a complex number x makes (*) true iff

$$s = x + x^{-1}$$ is a root of the $$Q(s) = s^2 + as + (b-2)$$

I see that that $$Q(x + x^{-1}) = \frac{p(x)}{x^2}$$

Then to prove the above do I then show that p(x) and Q((x + ^{-1}) shares roots?

Sincerely Yours
MM23
The onlly way a fraction can be 0 is if the numerator is 0. Isn't it obvious from $$Q(x + x^{-1}) = \frac{p(x)}{x^2}$$
that Q(x+ x-1)= 0 if and only if p(x)= 0?

1. What is a polynomial of degree 4?

A polynomial of degree 4 is an algebraic expression that contains a variable raised to the 4th power and has coefficients and constants. It can be written in the form ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are constants and x is the variable.

2. What are the roots of a polynomial of degree 4?

The roots of a polynomial of degree 4 are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation ax^4 + bx^3 + cx^2 + dx + e = 0.

3. How do you find the roots of a polynomial of degree 4?

To find the roots of a polynomial of degree 4, you can use the rational root theorem, synthetic division, or the quadratic formula. You can also use a graphing calculator to approximate the roots.

4. Can a polynomial of degree 4 have complex roots?

Yes, a polynomial of degree 4 can have complex roots. This means that the solutions to the equation ax^4 + bx^3 + cx^2 + dx + e = 0 can be complex numbers, which are numbers with a real and imaginary part. Complex roots usually come in pairs of complex conjugates.

5. How many roots does a polynomial of degree 4 have?

A polynomial of degree 4 can have up to 4 roots, but it is not guaranteed to have 4 distinct roots. It may have repeated roots or complex roots. The fundamental theorem of algebra states that a polynomial of degree n has n complex roots, counting multiplicity.

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