Problem 24 - 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. $$ x^{4}-2 x^{2}-16 x-15=0 $$

Step-by-Step Solution

Answer

The polynomial equation has one rational root at \(x = -1\). The remaining roots can be found by solving the cubic equation \(x^{3} - x^{2} + x - 13 = 0\) using synthetic division and factoring techniques.

Step 1: Rational Root Theorem

The Rational Root Theorem states that if the polynomial has a rational root written in lowest terms as \(p/q\), where \(p\) is a factor of the constant and \(q\) is a factor of the leading coefficient. So, first list all factors of the constant term, which is -15: \(\pm1, \pm3, \pm5, \pm15\). For this polynomial, the leading coefficient is 1, so the factors are \(\pm1\). Therefore, the possible rational roots are: \(\pm1, \pm3, \pm5, \pm15\)

Step 2: Synthetic Division

Synthetic division is a shortcut method of polynomial division, especially when dividing by a linear factor. We use synthetic division to test each of the possible roots until we find a root. Let's test 1: \[Synthetic division with 1 as root: \] \[1 | 1 0 -2 -16 -15 \] \[....+..1 1 -1 -17 \] \[......1 1 -1 -17 -32 \] The final result is -32, thus we know that 1 is not a root. Repeat this process with the remaining possible roots until we find an actual root.

Step 3: Find Actual Root

By repeating step 2, we find that -1 is indeed a root. Using synthetic division with -1 as root, we get the following: \[-1 | 1 0 -2 -16 -15 \] \[...-...1 -1 3 13 \] \[...1 -1 1 -13 0 \] The final entry in the last row is zero, so -1 is a root.

Step 4: Finding Remaining Roots

The quotient we get from the previous step if we throw out the 0 remainder, is a polynomial equation of degree 3. This equation represents the remaining roots of the original equation. We have: \(x^{3} - x^{2} + x - 13 = 0\). Now, this can further be factorized or solved using synthetic division again and factoring technique to find the remaining roots.

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing polynomials that is particularly useful when dividing by a linear factor. Unlike traditional polynomial long division, it requires fewer calculations and is easier to perform.
Let’s break down synthetic division:

• Choose a possible root based on the rational root theorem.
• Set up the division by writing down the coefficients of the polynomial.
• Draw a division bar and place the chosen root outside it.
• Bring down the leading coefficient to the bottom row.
• Multiply the chosen root by the number just written, placing the result under the next coefficient.
• Add, then continue this process until completion.
• If the remainder is zero, the chosen number is a root.

This process is repeated for other possible rational roots until an actual root is confirmed. Synthetic division helps in identifying actual roots and simplifying equations.

Polynomial Roots

Polynomial roots, or solutions, are values of the variable which satisfy the polynomial equation, making the equation equal to zero.
For a polynomial equation like the one in the exercise: \[x^{4} - 2x^{2} - 16x - 15 = 0\]the actual roots represent the values of \(x\) that satisfy this equation.
To find these roots, we test possible rational roots, which come from the Rational Root Theorem. Actual roots are confirmed through methods like synthetic division. Once a root is confirmed, the polynomial can be simplified. This continued simplification helps in determining all roots effectively. Roots can be real numbers or complex, and finding them is key to solving polynomial equations completely.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of its simplest polynomials, or factors. These factors are usually linear or quadratic expressions.
In the given exercise, once you've identified a root using synthetic division, you can express the original polynomial as a product of \[(x - r)\]and another lower degree polynomial, where \(r\) is a root.

• Use synthetic division to divide the polynomial by \((x - r)\).
• The quotient, without the remainder, is a factor of the polynomial.
• Repeat the process for higher-degree polynomials until the polynomial is fully factored.

Factorization simplifies complex polynomials and makes solving for roots easier. It's often used along with synthetic division and other algebraic techniques.

Polynomial Equations

Polynomial equations are mathematical expressions that express the equality between a polynomial and zero. They can have varying degrees, which tell us the highest power of the variable in the equation.
In the exercise, the equation given is a fourth-degree polynomial:\[x^{4} - 2x^{2} - 16x - 15 = 0\]To solve polynomial equations, identify possible rational roots, use synthetic division to find actual roots, and factor the polynomial to simplify it completely.

• Start with the Rational Root Theorem to list possible roots.
• Use synthetic division to eliminate non-roots and refine the equation.
• Simplify and factorize the polynomial to find all roots.

Solving polynomial equations involves understanding their structure, how to apply factorization techniques, and using tools like synthetic division effectively.