Problem 21 - Basic Technical Mathematics with Calculus

Question

Find the remaining roots of the given equations using synthetic division, given the roots indicated. $$x^{5}-3 x^{4}+4 x^{3}-4 x^{2}+3 x-1=0 \quad(1 \text { is a triple root })$$

Step-by-Step Solution

Answer

The remaining roots are \( i \) and \( -i \).

Step 1: Set Up the Equation for Synthetic Division

Since 1 is a root, and actually a triple root, we will perform synthetic division on the polynomial three times. Start by writing the coefficients of the polynomial: \(1, -3, 4, -4, 3, -1\).

Step 2: Perform First Synthetic Division

Using 1 as the root, perform synthetic division on the polynomial:1. Bring down the leading coefficient, 1.2. Multiply by the root (1) and add to the next coefficient: - \(1 \times 1 + (-3) = -2\)3. Repeat the process: - \(1 \times -2 + 4 = 2\) - \(1 \times 2 + (-4) = -2\) - \(1 \times -2 + 3 = 1\) - \(1 \times 1 + (-1) = 0\)The new coefficients are \(1, -2, 2, -2, 1\). The remainder is 0, confirming 1 as a root.

Step 3: Perform Second Synthetic Division

Repeat the synthetic division with the new set of coefficients \(1, -2, 2, -2, 1\):1. Bring down the 1.2. Multiply by the root (1) and add to the next coefficient: - \(1 \times 1 + (-2) = -1\)3. Continue with the same process: - \(1 \times -1 + 2 = 1\) - \(1 \times 1 + (-2) = -1\) - \(1 \times -1 + 1 = 0\)The new coefficients become \(1, -1, 1, -1\), with a remainder of 0.

Step 4: Perform Third Synthetic Division

Proceed with synthetic division on the new polynomial \(1, -1, 1, -1\):1. Bring down the 1.2. Multiply by the root (1) and add to the next coefficient: - \(1 \times 1 + (-1) = 0\)3. Continue: - \(1 \times 0 + 1 = 1\) - \(1 \times 1 + (-1) = 0\)The new coefficients are \(1, 0, 1\), and the remainder is 0.

Step 5: Identify Remaining Roots

Having reduced the polynomial to \(x^2 + 1 = 0\), solve for the remaining roots:1. Set the equation to zero: \(x^2 + 1 = 0\).2. Rearrange to get \(x^2 = -1\).3. Solve for \(x\): \(x = \pm i\).Hence, the non-real roots of the polynomial are \( i \) and \( -i \).

Key Concepts

Complex Roots

Complex roots are roots of a polynomial equation that are not real numbers. These appear when the polynomial has factors that do not cross the x-axis on the graph.
In the exercise, the simplified polynomial becomes \(x^2 + 1 = 0\). When we solve this, we end up with complex roots because \(x^2 = -1\) gives roots \(x = \pm i\). Here, \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\).

• Complex numbers have a real part and an imaginary part. For example, the number \(3 + 4i\) has 3 as the real part and 4 as the imaginary part.
• The complex roots \(i\) and \(-i\) are purely imaginary, meaning their real parts are zero.

Understanding complex roots is essential for fully grasping polynomials, especially when dealing with non-real solutions. They show up in many areas of mathematics, including engineering and physics.

Polynomial Equation

A polynomial equation is an expression involving a sum of powers of a variable. In this exercise, the original polynomial equation is of degree 5: \(x^{5}-3x^{4}+4x^{3}-4x^{2}+3x-1=0\). Each term has a coefficient and a power of \(x\).
Solving polynomial equations often involves finding the values of \(x\) (the roots) that make the equation equal zero. These roots can be real numbers or complex numbers.

• In our exercise, the polynomial is solvable using synthetic division, a handy tool when certain roots are already known.
• For any polynomial of degree \(n\), there will be \(n\) roots, which may include real and complex numbers, as well as repeated roots.
• Polynomials are critical in mathematics because they model a broad range of phenomena, and their roots can help to understand the behavior of the equations they represent.

By understanding polynomial equations, students will gain insights that are applicable in calculus, algebraic geometry, and more.

Root Multiplicity

Root multiplicity refers to the number of times a specific root appears in a polynomial equation. If a root "repeats" or occurs multiple times, it affects the shape and behavior of the polynomial graph.
In the given exercise, \(1\) is a triple root, meaning it is a root of the polynomial that occurs three times.

• Each time \(1\) is encountered as a root in synthetic division, it further reduces the polynomial's degree, simplifying the equation and revealing additional roots.
• A root with a higher multiplicity will cause the graph of the polynomial to "touch" the x-axis but not necessarily "cross" it at that point.
• The multiplicity of roots is crucial because it determines the polynomial’s factorization and helps uncover hidden relationships between the roots and coefficients.

Understanding root multiplicity is a vital skill in algebra, providing deeper insights into polynomial behavior and solutions.