Problem 57 - Algebra and Trigonometry

Question

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{5}-3 x^{4}+12 x^{3}-28 x^{2}+27 x-9 $$

Step-by-Step Solution

Answer

The zeros are \( x = 1 \) (multiplicity 3), \( x = 3i \), and \( x = -3i \).

Step 1: Understand the Polynomial

We are given the polynomial \( P(x) = x^5 - 3x^4 + 12x^3 - 28x^2 + 27x - 9 \). The task is to find all the zeros, which are the values of \( x \) that make \( P(x) = 0 \).

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial, in the form \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (-9) and \( q \) as a factor of the leading coefficient (1). Thus, the possible rational roots are \( \pm 1, \pm 3, \pm 9 \).

Step 3: Test Possible Rational Roots

Substitute each possible rational root into \( P(x) \) to check if it equals zero:- \( P(1) = 1 - 3 + 12 - 28 + 27 - 9 = 0 \) (so \( x = 1 \) is a root)- Testing other values, none yield zero, so \( x = 1 \) is initially confirmed.

Step 4: Synthetic Division

Use synthetic division to divide \( P(x) \) by \( (x - 1) \), since \( x = 1 \) is a confirmed root. This results in a quotient polynomial of degree 4: \( x^4 - 2x^3 + 10x^2 - 18x + 9 \).

Step 5: Repeat Steps 2-4 for Quotient Polynomial

Apply the Rational Root Theorem to the quotient polynomial. Test rational roots as before and find another root, \( x = 1 \). Perform synthetic division again to simplify further to \( x^3 - x^2 + 9x - 9 \).

Step 6: Solve Remaining Polynomial

Repeat the process for the cubic polynomial. After testing, find that \( x = 1 \) is a root once more. Divide using synthetic division to get \( x^2 + 9 \).

Step 7: Solve the Quadratic Equation

The quadratic \( x^2 + 9 \) is solved. Set \( x^2 + 9 = 0 \) which simplifies to \( x^2 = -9 \), giving solutions \( x = 3i \) and \( x = -3i \) using the square root property.

Step 8: List All Zeros

Combine the results of the previous steps. Repeated roots are \( x = 1 \) (with multiplicity 3), and the complex roots are \( x = 3i \) and \( x = -3i \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool for finding potential zeros of a polynomial with rational coefficients. It suggests that any rational solution to the equation \( P(x) = 0 \) can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers. Specifically, \( p \) is a factor of the constant term (here \(-9\)), and \( q \) is a factor of the leading coefficient (here \(1\)).

• For the polynomial \( P(x) = x^5 - 3x^4 + 12x^3 - 28x^2 + 27x - 9 \), potential rational roots are \( \pm 1, \pm 3, \pm 9 \).
• These potential roots are systematically checked to see if they satisfy \( P(x) = 0 \).

If a root results in the polynomial equating to zero, it's confirmed as a real root that the polynomial might cross or touch the x-axis at these values. This theorem gives a structured way to identify candidates for rational roots without wild guessing.

synthetic division

Synthetic division is a simplified form of polynomial division, specifically useful for dividing by linear factors like \( x - r \), where \( r \) is a root. This method saves time compared to long division and is particularly straightforward for evaluating polynomials at specific points.
To perform synthetic division:

• Write the coefficients of the polynomial in a row.
• Use the root (here \(1\)) to start the division.
• Drop the first coefficient straight down and multiply by the root, continuing the operation across all coefficients.

In our solution, synthetic division helped reduce the degree of the polynomial after confirming \( x = 1 \) is a root, simplifying to \( x^4 - 2x^3 + 10x^2 - 18x + 9 \). The process is repeated to factor the polynomial entirely.

complex numbers

Complex numbers come into play when dealing with roots of equations that don't yield real number solutions. They are expressed in the form \( a + bi \), where \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
In the given polynomial, after using synthetic division, we end up with \( x^2 + 9 \), a quadratic that doesn’t intersect the x-axis. Solving \( x^2 + 9 = 0 \) leads us to:

• Isolate \( x^2 \): \( x^2 = -9 \).
• Taking the square root gives \( x = \pm 3i \).

These solutions are complex roots. Complex roots often come in conjugate pairs, as seen here with \( x = 3i \) and \( x = -3i \). While they don't represent x-axis intersections, they're essential for understanding polynomial behaviors in the complex plane.

multiplicity of roots

The concept of the multiplicity of roots reveals how many times a particular root appears within a polynomial. When you perform polynomial factorization and encounter the same root repeatedly, it has a multiplicity greater than one.For \( P(x) = x^5 - 3x^4 + 12x^3 - 28x^2 + 27x - 9 \), we find that the root \( x = 1 \) repeatedly makes the polynomial zero. By using synthetic division multiple times, this root is confirmed to have a multiplicity of three.

• Each repetition of synthetic division confirms the multiplicity.
• A root's multiplicity indicates how the curve behaves at that point — for example, a root with odd multiplicity means the graph crosses the x-axis.

Recognizing the multiplicity is key to fully factoring the polynomial and understanding its graph. Every appearance of the same root reduces the polynomial's degree further, simplifying continued analysis.