Problem 9 - Basic Technical Mathematics with Calculus

Question

Find the remaining roots of the given equations using synthetic division, given the roots indicated. $$2 x^{3}+11 x^{2}+20 x+12=0 \quad\left(r_{1}=-\frac{3}{2}\right)$$

Step-by-Step Solution

Answer

The remaining roots are \(-2, -2\).

Step 1: Synthetic Division Setup

Write down the coefficients of the polynomial \(2, 11, 20, 12\) in sequence. Use the given root \(r_1 = -\frac{3}{2}\) as the divisor in synthetic division.

Step 2: Bring Down the First Coefficient

Bring down the leading coefficient, which is \(2\), to the bottom row.

Step 3: Multiply and Add Iteratively

1. Multiply \(2\) by \(-\frac{3}{2}\) and write the result under the next coefficient. Add to get the next number in the bottom row: \ \(2 \times -\frac{3}{2} = -3\), \, \(11 + (-3) = 8\).2. Repeat: \(8 \times -\frac{3}{2} = -12\), \ \(20 + (-12) = 8\).3. Repeat: \(8 \times -\frac{3}{2} = -12\), \ \(12 + (-12) = 0\). The remainder is \(0\).

Step 4: Write the Quotient Polynomial

The coefficients \(2, 8, 8\) from synthetic division give us the quadratic quotient \(2x^2 + 8x + 8 = 0\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula to find the roots of \(2x^2 + 8x + 8 = 0\): \ \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], where \(a = 2\), \(b = 8\), \(c = 8\).Calculate discriminant: \(8^2 - 4 \times 2 \times 8 = 64 - 64 = 0\).Since the discriminant is \(0\), there is one repeated root: \ \[x = \frac{-8}{4} = -2\].

Step 6: Confirm the Remaining Roots

The original cubic equation's root \(r_1 = -\frac{3}{2}\) and the double root from the quadratic \(x = -2\) confirm all roots: \ \(-\frac{3}{2}, -2, -2\).

Key Concepts

Polynomial Roots

When looking for polynomial roots, you're essentially finding the values where the polynomial becomes zero. For a cubic polynomial like \(2x^3 + 11x^2 + 20x + 12\), these roots are the solutions to the equation \(2x^3 + 11x^2 + 20x + 12 = 0\). Think of roots as the

• points on a graph where the curve touches the x-axis
• solutions to the polynomial equation

In our problem, we're given one root, \(r_1 = -\frac{3}{2}\). Using synthetic division with this root helps simplify the polynomial, making it easier to find the remaining roots. After applying synthetic division, the original polynomial is reduced to a quadratic form, which is easier to handle.

Quadratic Equation

The quadratic equation is a polynomial of degree 2, commonly written in the form \(ax^2 + bx + c = 0\). In the synthetic division process, we reduced the cubic polynomial to the quadratic \(2x^2 + 8x + 8 = 0\). Solving quadratic equations is generally straightforward using various methods:

• Factoring: If possible, express the quadratic in factor form \((px + q)(rx + s) = 0\).
• Quadratic Formula: A universal method given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: Transform the equation to make solving for \(x\) easier.

In this case, we used the quadratic formula to effortlessly find the root, because factoring was not viable and completing the square was unnecessary.

Discriminant Analysis

The discriminant in a quadratic equation \(ax^2 + bx + c = 0\) is the part under the square root in the quadratic formula: \(b^2 - 4ac\). This number tells you about the nature of the roots:

• Discriminant > 0: Two distinct real roots.
• Discriminant = 0: One repeated real root.
• Discriminant < 0: No real roots (the roots are complex).

In the quadratic \(2x^2 + 8x + 8\), the discriminant is \(64 - 64 = 0\), meaning we have one repeated real root \(x = -2\). This result confirms the presence of a double root at \(-2\), simplifying the original cubic's solutions to \(-\frac{3}{2}, -2, -2\). Analyzing the discriminant is a powerful tool as it directly reveals the nature of the roots without requiring complete solution calculation.