Problem 23 - Algebra and Trigonometry

Question

In Exercises \(17-24\) a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the quotient from part (b) to find the remaining roots and solve the equation. $$ x^{4}-2 x^{3}-5 x^{2}+8 x+4=0 $$

Step-by-Step Solution

Answer

The roots of the given polynomial equation \(x^{4}-2x^{3}-5x^{2}+8x+4=0\) are \(2, -1, 1 + \sqrt{2}, 1 - \sqrt{2}\).

Step 1: Listing All Possible Rational Roots

The Rational Root Theorem states that any possible rational root, p/q, of a polynomial is such that p is a factor of the constant term (in this case, 4), and q is a factor of the leading coefficient (here, 1). So, the possible rational roots of the polynomial are \(\pm 1, \pm 2, \pm 4\).

Step 2: Using Synthetic Division to Find an Actual Root

We then use synthetic division to test these possible roots. Starting with 1, we find that it is not a root. We repeat the process for -1, 2, -2, 4, and -4, until finding that 2 is an actual root of the polynomial.

Step 3: Using the Quotient to Find the Remaining Roots

With the use of synthetic division with 2 we get the quotient polynomial to be \(x^{3}-4x^{2}+x+2=0\). The roots of this cubic equation can be found by factoring or using the cubic formula. these roots will also be roots of the original polynomial equation. For instance, applying synthetic division again with \(x = -1\) and then factoring, we get the remaining roots as \(-1, 1 \pm \sqrt{2}\).

Key Concepts

Polynomial Equation

A polynomial equation is an expression of mathematical equality that involves variables, constants, and non-negative integer exponents. For example, the polynomial equation given in the exercise is \[x^{4} - 2x^{3} - 5x^{2} + 8x + 4 = 0\]. In this expression, the highest power of the variable, known as the degree, is 4.

Polynomials can have various degrees, and an equation of degree 4 is specifically called a quartic polynomial. Each term of the polynomial, like \( x^4 \), \( -2x^3 \), is called a monomial. These monomials are connected using addition or subtraction. To solve a polynomial equation like this,

• We often list all possible rational roots first.
• Then we find actual roots, using methods such as synthetic division.
• Finally, determine the remaining roots by factoring or applying further techniques.

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a binomial of the form \( x - r \). It is particularly useful for testing potential roots quickly and easily.

In our solution, synthetic division begins by testing potential roots found using the Rational Root Theorem, which suggests that possible rational roots of \[x^{4} - 2x^{3} - 5x^{2} + 8x + 4 = 0\] are \(\pm 1, \pm 2, \pm 4\). We systematically apply synthetic division with each possible root:

• If the remainder is zero, it confirms that the tested value is indeed a root of the original polynomial.
• When the remainder isn't zero, the tested value is not a root.
• Continuing this process for each candidate helps us find that the number 2 is a root.

The efficiency of synthetic division comes from its ability to compute without rewriting complex expressions, and only requires straightforward arithmetic to test roots.

Factoring

Factoring is a key algebraic process used to simplify polynomial expressions by writing them as a product of their constituents, known as factors. After identifying 2 as a root using synthetic division for our polynomial equation, we obtain the quotient \[x^{3} - 4x^{2} + x + 2\].

To find the remaining roots, we attempt to factor this cubic polynomial. Factoring involves trying to express it as a product of smaller polynomials that can potentially have degree 1 or 2. The steps in this process include:

• Checking easy-to-spot factors like grouping terms.
• Testing roots again through synthetic division if necessary.
• Continuing analysis with efficient algebraic manipulations to simplify the expression.

Upon successfully factoring or identifying additional roots through synthetic division again, our example reveals that the roots include \(-1\) and \(1 \pm \sqrt{2}\). These roots are solutions to both the initial and factored equation.

Cubic Equation

A cubic equation is a polynomial equation where the highest exponent of the variable is 3. It can be expressed in the form \[ax^{3} + bx^{2} + cx + d = 0\]. Our exercise generated such an equation, \[x^{3} - 4x^{2} + x + 2 = 0\], from the original polynomial division.

Cubic equations can be solved via factoring, synthetic division, or through formulas like Cardano's formula. In simpler cases,

• Look for rational roots using methods like the Rational Root Theorem.
• Factorise the cubic into a product of linear and quadratic factors, if possible.
• For more complex roots, employ algebraic techniques like completing the square or using known formulas.

After finding the rational root \( x = -1 \) using earlier methods, the remaining roots \( 1 \pm \sqrt{2} \) were derived through further algebraic factoring techniques. Understanding these processes allows one to efficiently solve and predict the behavior of cubic equations.