Problem 17 - Basic Technical Mathematics with Calculus

Question

Solve the given equations without using a calculator. $$D^{5}+D^{4}-9 D^{3}-5 D^{2}+16 D+12=0$$

Step-by-Step Solution

Answer

The roots of the equation include \( D = -1 \) and others from further factorization steps.

Step 1: Identify Possible Rational Roots

To find potential rational roots, we use the Rational Root Theorem. According to this theorem, potential rational roots are the factors of the constant term (12) divided by the factors of the leading coefficient (1). Thus, possible roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 2: Test Rational Roots

We need to evaluate the polynomial at these possible rational roots to find an actual root. Begin by testing \( D = 1 \). Substitute into the equation: \[ 1^5 + 1^4 - 9 \times 1^3 - 5 \times 1^2 + 16 \times 1 + 12 = 1 + 1 - 9 - 5 + 16 + 12 = 16 \] This is not zero, so \( D = 1 \) is not a root. Continue testing until a root is found.

Step 3: Verification of a Root

After testing several possibilities, try \( D = -1 \):\[ (-1)^5 + (-1)^4 - 9(-1)^3 - 5(-1)^2 + 16(-1) + 12 = -1 + 1 + 9 - 5 - 16 + 12 = 0 \]Thus, \( D = -1 \) is a root of the polynomial.

Step 4: Divide the Polynomial

Since \( D = -1 \) is a root, use synthetic division or polynomial division to divide the polynomial by \( D + 1 \). Perform the division step-by-step, noting the reminders as you deduce coefficients for the resulting polynomial.

Step 5: Solve the Reduced Polynomial

After dividing by \( D + 1 \), the remaining polynomial is \[ D^4 - 9D^3 - 5D^2 + 16D + 12 \]Repeat steps 1-4 to find further roots of this reduced polynomial, checking possible rational roots and dividing by factors that result in zero.

Step 6: Analyze Further Factorization

Continue dividing the polynomial until it is fully factorized or reduced into quadratic or linear terms. Solving any quadratic using the quadratic formula if necessary will give remaining roots. The fully factorized polynomial ultimately reveals all the rational and irrational roots.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a great tool used to find potential rational solutions of polynomial equations. It acts like a treasure map, guiding you to possible roots of a polynomial without guessing blindly. According to the theorem, if a polynomial has a rational root, that root could be expressed as a fraction \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term
• \( q \) is a factor of the leading coefficient

So, in our exercise where the polynomial is \( D^5 + D^4 - 9D^3 - 5D^2 + 16D + 12 \), the constant term is 12, and the leading coefficient is 1. Therefore, the potential rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \). By evaluating these potential roots, we can quickly search for actual roots of the polynomial.

Synthetic Division

Once a potential root is identified using the Rational Root Theorem, the next step is to verify and utilize this root with the help of synthetic division. Synthetic division is a simplified form of polynomial long division, often used when dividing by linear factors such as \( (D - c) \).To perform synthetic division:

• Write down the coefficients of the polynomial.
• Place the identified root \( c \) outside of an upside-down division symbol.
• Carry down the first coefficient, then multiply it by the root and add to the next coefficient.
• Continue this process across, repeating multiply and add, until complete.
• If the final value is zero, \( c \) is indeed a root, and you have found new coefficients for the quotient polynomial.

This method not only checks the correctness of the root but also simplifies the polynomial, making it smaller and more manageable for further operations.

Polynomial Factorization

When working with polynomials, finding its factors is crucial to discovering its roots. Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. Here's how you do it:

• Use identified roots to repeatedly apply synthetic division, breaking the polynomial into smaller factors.
• Continue testing and re-evaluating potential roots using the Rational Root Theorem.
• Repeat the process of dividing until the polynomial is fully decomposed into irreducible quadratic or linear factors.

Through complete factorization, you can easily extract all rational and potentially some irrational roots of the polynomial, leading to a comprehensive solution of the polynomial equation.

Quadratic Formula

In some cases, polynomial factorization will result in a quadratic equation, which can occasionally be challenging or impossible to factor further by simple methods. This is where the Quadratic Formula comes in handy. The formula allows you to find the roots of any quadratic polynomial, expressed as:
\[ D = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For a quadratic polynomial \( aD^2 + bD + c \), simply substitute in the respective coefficients.Important points to remember:

• The expression under the square root, \( b^2 - 4ac \), is known as the discriminant. It tells you about the nature of the roots.
• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, the roots are complex and not real numbers.

The Quadratic Formula is a powerful tool to ensure you don’t miss any crucial solutions when dealing with quadratic factors.