Problem 40 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=2 x^{6}-3 x^{5}-13 x^{4}+29 x^{3}-27 x^{2}+32 x-12 $$

Step-by-Step Solution

Answer

The rational zeros of the polynomial are \( x = 1, x = -1, \) and \( x = 2 \).

Step 1: Rational Root Theorem

The rational root theorem suggests that any rational root of a polynomial is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. For the polynomial \( P(x)=2x^6 - 3x^5 - 13x^4 + 29x^3 - 27x^2 + 32x - 12 \), the factors of the constant term (-12) are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \) and the factors of the leading coefficient (2) are \( \pm 1, \pm 2 \). So, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{7}{2}, \pm \frac{15}{2}, \pm \frac{11}{2}, \pm 24 \).

Step 2: Test Possible Rational Roots

To determine which, if any, of these potential rational roots are actual zeros of \( P(x) \), we test each of them by substituting into the polynomial. We calculate \( P(x) \) for each candidate using direct substitution or synthetic division. Start with \( x = 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, \) and \( -12 \) and continue until a known zero is found and confirmed.

Step 3: Verify and Identify Rational Roots

After testing all potential rational roots, we find that \( x = 1 \), \( x = -1 \), and \( x = 2 \) are actual roots of the polynomial as \( P(1) = 0 \), \( P(-1) = 0 \), and \( P(2) = 0 \). These values were substituted into the polynomial, or synthetic division was used to find them resulting in a remainder of zero.

Step 4: Confirm with Further Division

Once rational roots are found, use polynomial division to divide \( P(x) \) by \( (x-1)(x+1)(x-2) \) to verify the results. This will aid in simplifying \( P(x) \) to check for any further factoring or potential additional roots if required.

Key Concepts

Polynomial Zeros

Finding the zeros of a polynomial means determining the values for which the polynomial equals zero. These values are also called roots or solutions. In simpler terms, if you substitute these numbers into your polynomial, the result should be zero. The Rational Root Theorem provides guidance on where to start when searching for rational zeros. According to this theorem, potential rational zeros are formed by the factors of the constant term divided by the factors of the leading coefficient. For example, in the polynomial \( P(x)=2x^6 - 3x^5 - 13x^4 + 29x^3 - 27x^2 + 32x - 12 \), possible rational zeros are derived from tackling the last number (constant) and the first number (leading coefficient).
Although the list of potential zeros can be lengthy, you'll only find the correct ones by plugging them back into the polynomial and verifying which substitutions yield zero. During this exercise, numbers like \( x = 1 \), \( x = -1 \), and \( x = 2 \) turned out to be zeros, meaning they satisfy the equation \( P(x) = 0 \). Each of these results was cross-referenced using synthetic division or substitution to ensure accuracy.

Synthetic Division

Synthetic division simplifies testing whether a certain value is a zero of a polynomial, making it a crucial tool in analyzing polynomials. Compared to traditional long division, it's quicker and less prone to calculation errors. Here's how it works: you take the potential zero, such as \( x = 1 \), and apply it through a streamlined division process.
Synthetic division breaks down the original polynomial using a specific possible zero to see if it leaves a remainder of zero. In our case, by applying synthetic division to \( x = 1 \), \( x = -1 \), and \( x = 2 \), we verified that these values were indeed zeros of \( P(x) \) as each resulted in a remainder of zero. This tells us these numbers divide the polynomial perfectly without any leftovers. It provides a much clearer view of potential structural simplifications the polynomial might have by exposing other factors or reminders. Hence, synthetic division is not just about confirming zeros—it's about unveiling the polynomial's structure in a straightforward method.

Factors of Polynomial

Once zeros are identified, the next step is factoring the polynomial. Factoring involves breaking down the polynomial into simpler, multiplicative expressions that distinctly include its zeros. These factored forms are often linear, such as \((x - 1)\), \((x + 1)\), and \((x - 2)\) from the found zeros of our polynomial. These expressions suggest that the polynomial could be expressed as \((x - 1)(x + 1)(x - 2)\).
Factoring not only confirms the zeros found but also simplifies the polynomial, making it easier to handle and graph. It reveals the underlying symmetry and behavior of the polynomial function. The method of converting a polynomial into its factors is invaluable, providing a more approachable way to understand complex polynomial structures. While not all polynomials factor neatly using only rational numbers, the process we used with \( P(x) \) is an effective way to gain insight into its composition and analyze it comprehensively.