Problem 36 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=6 x^{4}-7 x^{3}-12 x^{2}+3 x+2 $$

Step-by-Step Solution

Answer

The rational zeros are \( x = \frac{1}{2}, x = \frac{2}{3}, x = -1, \) and \( x = 1 \).

Step 1: Identify Possible Rational Zeros

According to the Rational Root Theorem, possible rational zeros of the polynomial \( P(x) = 6x^4 - 7x^3 - 12x^2 + 3x + 2 \) are of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term 2, and \( q \) is a factor of the leading coefficient 6. List the factors: \( p: \pm 1, \pm 2 \) \( q: \pm 1, \pm 2, \pm 3, \pm 6 \).Thus, the possible rational zeros are \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{6}\).

Step 2: Test Possible Zeros Using Synthetic Division

Test each possible rational zero by performing synthetic division. If the remainder is 0, then you have found a rational zero.Start with \( x = 1 \): after performing synthetic division with \( P(x) \), the remainder is not zero.Proceed with \( x = -1 \) (and repeat for each candidate) until a zero remainder is found.Repeat this process and discover that \( x = \frac{1}{2} \) is a root.

Step 3: Polynomial Division

Once a zero \( x = \frac{1}{2} \) is found, use the result from synthetic division to divide the polynomial. This division gives a reduced polynomial. In this case, divide \( P(x) \) by \( (2x - 1) \) to get a new polynomial.\( P(x) = (2x-1)(3x^3 - 2x^2 - 7x - 2) \).

Step 4: Repeat the Process on the Quotient

Apply the Rational Root Theorem again on the quotient \( 3x^3 - 2x^2 - 7x - 2 \). List possible rational zeros as before, and test them using synthetic division. Continue the process until further zeros are found, making sure each test leaves a zero remainder. After testing, find additional zeros as \( x = \frac{2}{3} \), \( x = -1 \), and another \( x = 1 \).

Step 5: Confirm All Zeros

Having found all zeros \( x = \frac{1}{2}, x = \frac{2}{3}, x = -1, x = 1 \), perform synthetic division or direct substitution to confirm that each zero gives a remainder of 0 when substituted back into the polynomial.

Key Concepts

Polynomial Division

Polynomial division is similar to long division in arithmetic, but instead of numbers, you're working with polynomials. The objective is to divide one polynomial by another, usually to simplify the polynomial or to find factors. This process is crucial when you want to break down complicated polynomial equations into simpler parts.

When performing polynomial division, the "dividend" is the polynomial you're dividing, and the "divisor" is the polynomial by which you're dividing. Typically, this involves arranging the polynomials in descending order of their degrees. The result of the division gives you a "quotient" and sometimes a "remainder."

• Quotient: The polynomial obtained from the division.
• Remainder: What's left after dividing. It should have a degree lower than the divisor.

Polynomial division is made more efficient by using synthetic division, a shortcut method particularly useful for dividing polynomials by linear expressions of the form \(ax - b\).

Synthetic Division

Synthetic division is a simplified form of polynomial division used when dividing by a linear factor. It's a staple technique in algebra due to its efficiency. Unlike the full polynomial division, synthetic division involves fewer steps and reduces the chance of errors.

To use synthetic division, you follow a straightforward series of steps:

• List the coefficients of the polynomial in descending order.
• Use the root of the divisor \(a\) for the division steps.
• Perform successive arithmetic operations, avoiding full polynomial calculations.

An important advantage of synthetic division is that it not only checks for rational zeros but also provides a way to factor the polynomial into simpler components. This helps both in solving polynomial equations and simplifying expressions. If the remainder at the end of synthetic division is zero, the divisor is indeed a factor of the polynomial.

Rational Zeros

Rational zeros of a polynomial refer to the x-values where the polynomial equals zero, and these zeros are rational numbers. A rational number is anything that can be expressed as a fraction \( \frac{p}{q} \), where \(p\) and \(q\) are integers.

The Rational Root Theorem aids in finding these zeros. It states that if \( \frac{p}{q} \) is a zero of the polynomial \(P(x)\), then \(p\) must be a factor of the constant term, and \(q\) must be a factor of the leading coefficient.

By listing the possible factors of these terms, you can determine potential rational zeros. Though not all will work, testing each result against the polynomial will uncover actual rational solutions. The presence of rational zeros can help simplify a polynomial equation, making them easier to solve or factor.

Polynomial Roots

Polynomial roots are the values of \(x\) where the polynomial equation \(P(x) = 0\) holds true. These might be integers, fractions, or even irrational numbers, but they are the critical components in solving or factoring polynomial equations.

To find these roots, especially the rational ones, you begin by determining possible candidates using the Rational Root Theorem. Testing these candidates with synthetic division or direct substitution helps confirm actual roots.

Identifying all roots is vital as they reveal much about the behavior of the polynomial function:

• Intersections with the x-axis
• Symmetry and overall shape of the graph
• Possible factoring strategies

The roots not only solve polynomial equations but also contribute to understanding the algebraic structure, allowing for deeper analysis and application in further mathematical exploration.