Problem 20 - Basic Technical Mathematics with Calculus

Question

Solve the given equations without using a calculator. $$2 x^{5}+5 x^{4}-4 x^{3}-19 x^{2}-16 x=4$$

Step-by-Step Solution

Answer

Reorganize the equation and test possible roots using synthetic division. Continue factoring by testing roots until simplified.

Step 1: Reorganize the Equation

First, move all terms to one side of the equation so it equals zero. We begin with \(2x^5 + 5x^4 - 4x^3 - 19x^2 - 16x = 4\). By subtracting 4 from both sides, we get: \[2x^5 + 5x^4 - 4x^3 - 19x^2 - 16x - 4 = 0\]

Step 2: Factor Out Common Terms

Look for a common factor in the polynomial. Notice that there is no greatest common factor for all terms, so we attempt to factor by grouping or testing possible rational roots.

Step 3: Use Synthetic Division for Potential Roots

To find potential roots, use the Rational Root Theorem, which suggests trying the factors of the constant term (-4) over the factors of the leading coefficient (2). This means we test possible roots: \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{2} \). Perform synthetic division to see if any of these roots reduce the polynomial to a quotient with no remainder.

Step 4: Test a Root Using Synthetic Division

Start with \(x = 1\): Use synthetic division on the polynomial. The coefficients are \([2, 5, -4, -19, -16, -4]\). After performing synthetic division, if the remainder is 0, then \(x = 1\) is a root. You can find similar results with other potential rational roots until reaching a solution.

Step 5: Factor the Equation Further

Once a root is found (e.g., \(x = 1\) is not a root based on synthetic division), test more values or use methods like polynomial long division with any found root. Simplify the polynomial until fully factored, or identify another root to continue.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool when working with polynomial equations. It helps you determine possible rational roots of a polynomial by focusing on the factors. If you have a polynomial like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\), the theorem suggests that any potential rational root, \(\frac{p}{q}\), will have \(p\) as a factor of the constant term \(a_0\) and \(q\) as a factor of the leading coefficient \(a_n\).

This means for our example, \(2x^5 + 5x^4 - 4x^3 - 19x^2 - 16x - 4 = 0\), the constant term is \(-4\) and the leading coefficient is \(2\). We should then consider the factors of these numbers. The possible values for \(p\) are \(\pm 1, \pm 2, \pm 4\), and \(q\) has \(\pm 1, \pm 2\). This means potential roots to test are \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}\). This theorem helps narrow down the roots you have to check, saving time and effort.

It's important to test each potential root using synthetic division until you find one that leaves no remainder, indicating a true root.

Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically designed to handle divisions by expressions like \((x - c)\). It is quicker and less cumbersome than the traditional long division method. For synthetic division, you only need the coefficients of the polynomial and the value of the possible root derived from the Rational Root Theorem.
Here's how it works:

• Write down the coefficients of the polynomial.
• Choose a potential root from your list and place it to the left of these coefficients.
• Drop the leading coefficient to the bottom row.
• Multiply this number by the potential root and write it under the next coefficient.
• Add the columns and repeat the multiplication with the result, writing each new product under the subsequent coefficient until you reach the end.

If the last value (remainder) is zero, then the number you tested is a root. This whole process simplifies checking each potential root significantly, helping to identify and isolate valid roots quickly.

Factoring Polynomials

Once a root is found via synthetic division that leaves no remainder, you can reduce the original polynomial by removing this root and factoring the smaller polynomial further. Factoring simplifies the original expression into a product of simpler polynomials, each representing a potential root or remaining factor.

Factorization may reveal a polynomial as a combination of linear factors or quadratic expressions, depending on the degree and nature of the roots:

• **Linear factors:** These appear as expressions like \((x - r)\) where \(r\) is a root.
• **Quadratic factors:** If no further linear factors can be found through rational root testing, quadratics might indicate irreducible expressions using real numbers.

By repeatedly applying synthetic division and factoring techniques, you can break down higher-degree polynomials into completely factored forms, allowing full understanding of all roots, including multiplicity when applicable. This structured breakdown makes solving such polynomial equations more manageable.