Problem 27 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$P(x)=x^{4}+8 x^{3}+24 x^{2}+32 x+16$$

Step-by-Step Solution

Answer

The only rational zero is \( x = -2 \), and it is a repeated zero.

Step 1: Identify the Constant and Leading Coefficient

The given polynomial is \( P(x) = x^4 + 8x^3 + 24x^2 + 32x + 16 \). Here, the constant term is \(16\) and the leading coefficient (the coefficient of the highest degree term, \(x^4\)) is \(1\).

Step 2: List Possible Rational Zeros

According to the Rational Root Theorem, any rational zero, \( \frac{p}{q} \), of a polynomial is such that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. The factors of \(16\) are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \). Since the leading coefficient is \(1\), \( q \) will be \( \pm 1 \).

Step 3: Test Possible Zeros

Use synthetic division or direct substitution to test the possible rational zeros \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \). Start with \( x = -2 \) and synthetic division shows it is a zero. Continue synthetic division with \( x = -2 \) on \( x^3 + 6x^2 + 12x + 8 \), yielding a zero again. The remaining polynomial is \( x^2 + 4x + 4 \).

Step 4: Solve the Quadratic Polynomial

Factor the quadratic polynomial \( x^2 + 4x + 4 \), resulting in \( (x+2)^2 \). Thus, \( x = -2 \) is a repeated zero.

Step 5: Verify the Zeros

Since \( x = -2 \) is a zero found repeatedly, plug back into \( P(x) \) to confirm zero result for verification. Therefore, \( P(-2) = 0 \).

Key Concepts

Rational zeros

When dealing with polynomials, finding the rational zeros (also known as rational roots) is a common task. According to the Rational Root Theorem, any rational zero can be confirmed if it takes the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
For our polynomial \( P(x) = x^4 + 8x^3 + 24x^2 + 32x + 16 \):

• The constant term is \(16\).
• The leading coefficient is \(1\).

The potential rational zeros are derived from the factors of these numbers. In this case, these would be \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \). These values are the possible candidates to test for actual zeros of the polynomial.
Always remember to use the Rational Root Theorem as a starting guide to list possible solutions.

Synthetic division

Synthetic division is a streamlined method for dividing polynomials when you are testing potential zeros. This method simplifies the process by avoiding long polynomial division, focusing on coefficients and constants.
Here’s how synthetic division is applied to test the possible zeros of \( P(x) \): First, you write down the coefficients of the polynomial: \(1, 8, 24, 32, 16\). Then, you select \( x = -2 \) (one of the potential rational zeros) as your central divisor.

• Begin by bringing down the leading coefficient (1) to start the synthetic division below the line.
• Multiply by \(-2\) and add to the next coefficient, continuing this process across the line.
• Repeat until you've processed all coefficients.

If the result is a zero remainder, \( x = -2 \) is indeed a zero of the polynomial.
This step needs repeating for any new polynomial generated until you arrive at an irreducible one for potential further factoring.

Polynomial factoring

Factoring polynomials is essential to simplify expressions and find zeros. Once potential zeros are validated using synthetic division, you can focus on factoring what's left.
For example, after confirming that \( x = -2 \) is a zero of \( P(x) \), synthetic division simplifies the polynomial to \( x^3 + 6x^2 + 12x + 8 \). Reapplying this technique simplifies it further to \( x^2 + 4x + 4 \).
The key is breaking down the polynomial into factors over its degree:

• First, use any zero found to reduce the polynomial.
• Apply factoring techniques (e.g., recognize perfect square trinomials).

Success in this leads to a more manageable expression that offers insight into further zeros, such as identifying the repeated factor \( (x+2)^2 \).

Quadratic polynomial

A quadratic polynomial is a polynomial of degree 2. In this context, it appears in the final stages of factoring higher-degree polynomials down to simpler parts.
The simplified part of our initial polynomial \( P(x) \) is \( x^2 + 4x + 4 \). Recognizing this as a perfect square trinomial is valuable. It factors into \( (x+2)^2 \), indicating it’s simply a repeated zero.
Key points about quadratic polynomials:

• They take the form \( ax^2 + bx + c \).
• They are crucial in finding and confirming all polynomial zeros.
• Solving quadratics can involve factoring, completing the square, or using the quadratic formula.

Understanding these forms helps confirm solutions and verifies the polynomial’s structure when simplified to its core components.