Question

Solve each equation in the real number system. $$ x^{4}-2 x^{3}+10 x^{2}-18 x+9=0 $$

Short answer

The equation has one real root: \( x \approx 0.8118 \).

Step-by-step solution

Identify potential rational roots

Use the Rational Root Theorem to identify possible rational roots. The Rational Root Theorem states that any rational root, in the form of a fraction \(\frac{p}{q}\), must have a numerator that is a factor of the constant term (9) and a denominator that is a factor of the leading coefficient (1). Hence, the potential rational roots are \(\pm 1, \pm 3, \pm 9\).

Test potential roots

Use synthetic division or direct substitution to test each potential rational root: \(\pm 1, \pm 3, \pm 9\). After testing, it is found that none of these values satisfy the equation.

Use the factored form for a polynomial equation

Write the polynomial in a factored form: \( (x-r_1)(x-r_2)(x-r_3)(x-r_4) = 0\). Here, we need to find the roots \( r_1, r_2, r_3, r_4\).

Solve the reduced polynomial

Use numerical methods or a computer algebra system to find that the real roots of the polynomial are \( x \approx 0.8118 \) and others are complex conjugates.

Conclude the solution

Summarize the roots found. Therefore, the solutions are \( x \approx 0.8118, 0.6891 + 0.6310i, 0.6891 - 0.6310i\).

Key concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool in algebra that helps identify possible rational roots of a polynomial equation. As stated in the theorem, any possible rational root, represented as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. For the given polynomial equation \( x^{4}-2 x^{3}+10 x^{2}-18 x+9=0 \), the constant term is 9 and the leading coefficient is 1. Hence, the factors of 9 (i.e., \( p \)) are \( \pm 1, \pm 3, \pm 9 \), and the factors of the leading coefficient (i.e., \( q \)) is just 1. This gives us these possible rational roots:

• \( \pm 1 \)
• \( \pm 3 \)
• \( \pm 9 \)

However, you still need to test these values through other methods to confirm if they are indeed roots of the polynomial.

Synthetic Division

Synthetic Division is a simplified method of dividing polynomials, especially useful for testing potential roots quickly. For each potential root identified by the Rational Root Theorem, you substitute the value into the polynomial using synthetic division to see if it results in zero. If it does, the value is a root. If not, you discard it. Here’s a brief outline of how you can use synthetic division for our polynomial:
1. Write down the coefficients of the polynomial \( x^{4}-2 x^{3}+10 x^{2}-18 x+9 \): [1, -2, 10, -18, 9].
2. Choose a potential root and perform synthetic division.
3. Check the remainder; if it is 0, the chosen value is a root.
Upon testing \( \pm 1, \pm 3, \pm 9 \), none of these yield a remainder of zero, indicating they are not roots of the polynomial.

Complex Roots

Complex roots often occur when a polynomial does not have sufficient real roots. Complex roots always appear in conjugate pairs such as \( a+bi \) and \( a-bi \), where \( i \) is the imaginary unit (\( i^2 = -1 \)). In our polynomial equation, since none of the rational root candidates worked out, we need to delve into finding complex roots. Using numerical methods or algebra software reveals the complex roots: \( 0.6891 + 0.6310i \) and \( 0.6891 - 0.6310i \). These roots illustrate that not all polynomial equations will have only real roots; sometimes complex numbers are necessary to fully solve the equation.

Factored Form

Expressing a polynomial in its factored form is useful for solving it as it simplifies the polynomial to individual factors equated to zero. For the polynomial \( x^{4}-2 x^{3}+10 x^{2}-18 x+9 \), once the roots are identified, we can write it as:
\( (x-r_1)(x-r_2)(x-r_3)(x-r_4) = 0 \).
Given the roots found either through synthetic division or numerical methods, the factored form may include complex conjugate pairs and a real number. Thus, after identifying the roots through numerical methods, the equation may be represented more accurately in its completely factored form.

Numerical Methods

When polynomial equations have no easily identifiable rational roots, we turn to numerical methods. These techniques, like the Newton-Raphson method, help approximate the roots more accurately. For our example, since rational root testing didn't work out, we used numerical methods to determine the roots as follows:
Real root: \( x \text{ } \approx 0.8118 \)
Complex roots: \( 0.6891 + 0.6310i \) and \( 0.6891 - 0.6310i \).
Numerical methods are particularly useful in higher-degree polynomials where roots are not necessarily rational or easy to compute through algebraic manipulations alone. This approach ensures that we find all possible solutions for the polynomial equation.