Problem 404 - Precalculus

Question

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ 3 x^{3}-x^{2}-11 x-6=0 $$

Step-by-Step Solution

Answer

Real zeros are \(x = -1\) and \(x = \frac{7}{3}\).

Step 1: Understand the Rational Zero Theorem

The Rational Zero Theorem states that any rational zero of a polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + ... + a_0 = 0\) is a fraction \(\frac{p}{q}\) where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). For the equation \(3x^3-x^2-11x-6=0\), the constant term \(a_0\) is \(-6\) and the leading coefficient \(a_n\) is \(3\).

Step 2: List Factors of the Constant Term and Leading Coefficient

List all factors of the constant term \(-6\), which are \(\pm 1, \pm 2, \pm 3, \pm 6\). List all factors of the leading coefficient \(3\), which are \(\pm 1, \pm 3\).

Step 3: Determine Possible Rational Zeros

Using the factors from Step 2, create a list of all possible rational zeros using \(\frac{p}{q}\). This results in the possible zeros \(\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}\).

Step 4: Test Possible Rational Zeros

Substitute each possible zero into the polynomial equation and evaluate to determine the zeros. Start with whole numbers: for \(x = -1\), \(3(-1)^3 - (-1)^2 - 11(-1) - 6 = 0\); so \(x = -1\) is a zero.

Step 5: Perform Synthetic Division

Use synthetic division with \(x = -1\) as zero:\[\begin{array}{r|rrrr} -1 & 3 & -1 & -11 & -6 \ & & -3 & 4 & 7 \\hline & 3 & -4 & -7 & 0 \\end{array}\]The quotient is the polynomial \(3x^2 - 4x - 7\).

Step 6: Solve the Quotient Polynomial

Solve for the remaining zeros in \(3x^2 - 4x - 7\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here, \(a = 3\), \(b = -4\), \(c = -7\). Substitute the values:\[x = \frac{4 \pm \sqrt{(-4)^2 - 4 \times 3 \times (-7)}}{6}\]\[x = \frac{4 \pm \sqrt{16 + 84}}{6}\]\[x = \frac{4 \pm \sqrt{100}}{6}\]\[x = \frac{4 \pm 10}{6}\].

Step 7: Calculate and Verify Remaining Zeros

Calculate \(x = \frac{14}{6} = \frac{7}{3}\) and \(x = \frac{-6}{6} = -1\). Since \(x = -1\) was found before, the real zeros are \(x = -1\) and \(x = \frac{7}{3}\).

Key Concepts

Polynomial Zeros

Polynomial zeros are values of \(x\) for which the polynomial equation evaluates to zero. These are the solutions or roots of the polynomial equation. Finding the zeros is a crucial task in algebra, as it provides vital information about the polynomial's behavior and graph.
To find the zeros of a polynomial like \(3x^3 - x^2 - 11x - 6 = 0\), we use the Rational Zero Theorem as a starting point. This theorem helps generate a list of possible rational zeros. By systematically testing these values, either through direct substitution or tools like synthetic division, the real zeros can be determined.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It's a quick alternative to the standard long division for polynomials and is especially useful in identifying zeros and factoring polynomials.
When using synthetic division with our polynomial and zero candidate \(x = -1\), the process involves setting up a division tableau and performing simple arithmetic operations. This results in a smaller polynomial, often making it easier to further identify remaining roots or factors. For instance, in our solution, using synthetic division for \(x = -1\) results in the reduced polynomial \(3x^2 - 4x - 7\). This can be further analyzed for additional zeros.

Quadratic Formula

The quadratic formula is a powerful algebraic tool used to find the zeros of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our exercise, after applying synthetic division, we arrive at the quadratic \(3x^2 - 4x - 7\). To find its zeros, we substitute \(a = 3\), \(b = -4\), \(c = -7\) into the quadratic formula. By solving the resulting expressions,

• \(x = \frac{14}{6} = \frac{7}{3}\)
• \(x = -1\)

These calculations reveal the remaining real zeros, helping fully solve the polynomial equation.

Factorization of Polynomials

Factorization of polynomials involves breaking down a polynomial into a product of simpler polynomials. This is analogous to expressing a number as a product of its prime factors.
For our particular polynomial equation, once the real zeros \(x = -1\) and \(x = \frac{7}{3}\) are determined, the polynomial can be expressed as a factored form. The full factorization would include these zeros, simplifying the expression into products that readily reveal its roots or solutions. Factorization is vital for solving polynomial equations, as it provides insights into their structure and properties.