Problem 54 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=3 x^{3}-5 x^{2}-8 x-2 $$

Step-by-Step Solution

Answer

The polynomial \( P(x) = 3x^3 - 5x^2 - 8x - 2 \) has no rational roots.

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial is a fraction \( \frac{p}{q} \) where \( p \) is a factor of the constant term (-2) and \( q \) is a factor of the leading coefficient (3). The possible rational roots are \( \pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3} \). We will test these to determine if any are roots of the polynomial.

Step 2: Test Possible Rational Roots

We'll begin by substituting these possible roots into the polynomial \( P(x) \):- \( P(1) = 3(1)^3 - 5(1)^2 - 8(1) - 2 = -12 \)- \( P(-1) = 3(-1)^3 - 5(-1)^2 - 8(-1) - 2 = -12 \)- \( P(2) = 3(2)^3 - 5(2)^2 - 8(2) - 2 = -10 \)- \( P(-2) = 3(-2)^3 - 5(-2)^2 - 8(-2) - 2 = -54 \)- \( P\left(\frac{1}{3}\right) \approx -6.07 \) - \( P\left(-\frac{1}{3}\right) \approx -2.93 \) - \( P\left(\frac{2}{3}\right) \approx -5.37 \) - \( P\left(-\frac{2}{3}\right) \approx -0.85 \).None of these tested values equal zero, indicating that the polynomial does not have any rational roots.

Step 3: Use Synthetic Division and Quadratic Formula

Since none of the rational roots worked, assume a possible simplification by using the factors obtained from similar calculations or prefixed assumptions (if algebra manipulation led us) to reduce finding other real roots. However, traditionally, one would either: 1. Try re-evaluating polynomial manipulations (numerically or graphically) to find approximate real zeros like re-calibrating synthetic trial values or, 2. Assume using aids or trial graphical conclusion a candidate root that we didn't find with small precision discrepancies, follow with subsequent synthetic division for easier reduction. - For simplicity here, it’s common, though applying digital tools, precise recursion can solve tri-variable evaluations whereby use of algorithms iteratively fits real zeros eg. opt beyond human selection.

Step 4: Graphical Interpretation or Approximate Tool Use

Using graphing calculators or software like Desmos can help visualize the polynomial to identify approximate roots not easily found analytically. Visual inspection can suggest approximate numerical solutions like at points where changes occur between negative and positive values, significant relative to trends seen or intermittent zero-cross lines and vertices, refined through iterative solver inputs. Here the suggested finding exhibit known switches and where intervals binary crossings scoping focused digital iterative trials (between x-values noted such as -1, 0, or 1) refine approximate values checked in phrase or practice root-verifier tools such 'Newton’s method’ on programming basics or aides.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool used to find possible rational roots of a polynomial. It tells us that any potential rational root, \( \frac{p}{q} \), must be a fraction where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. For our polynomial \( P(x) = 3x^3 - 5x^2 - 8x - 2 \), the constant term is -2, and its factors are \( \pm 1, \pm 2 \). The leading coefficient is 3, and its factors are \( \pm 1, \pm 3 \). Thus, the possible rational roots are \( \pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3} \).
By checking these through substitution into the polynomial, we found none of these values result in zero, indicating there are no rational roots. This means we need other methods to find the real zeros, such as synthetic division or graphing.

Synthetic Division

Synthetic division is a streamlined way of dividing a polynomial by a linear factor and is very useful in checking potential roots of polynomials. It involves fewer steps compared to the standard division of polynomials, making it quicker and easier to perform.
In our situation, even though none of the potential rational roots worked out, synthetic division could still be useful if we assumed a candidate prompt based on other findings or estimations.
Here's a simple process for synthetic division:

• Write down the coefficients of the polynomial.
• Choose a possible root, generally one suspected from graphical tools or previous attempts.
• Perform the synthetic division and observe the remainder. A remainder of zero confirms it's a root.

This process helps break down the polynomial into more manageable factors or validate roots.

Graphing Polynomials

Graphing polynomials is a powerful visual method to observe the behavior of functions and to find where they intersect the x-axis, indicating real zeros. Using digital tools like graphing calculators or software platforms like Desmos, students can directly visualize the polynomial's curve.
By plotting \( P(x) = 3x^3 - 5x^2 - 8x - 2 \), we'll look for points where the curve crosses the x-axis. These points suggest the potential real roots of the polynomial. The graph may show approximate intersections that can be further examined.

• Where the curve crosses the x-axis, it suggests a real zero.
• Total crossings on the graph hint at real zero counts.
• Always check these approximations with numerical methods for precision.

This graphical insight complements analytical methods, ensuring a more profound understanding of potential roots.

Quadratic Formula

When dealing with quadratics or when polynomial factors can be reduced into quadratics, the quadratic formula becomes invaluable: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For cubics like ours that resist simple factorization, synthetic division might reduce the polynomial to a quadratic form, which can then be solved readily with this formula.
If a cubic polynomial is split into a linear factor and a quadratic portion, the quadratic formula can dissect this part to find its real or complex roots. This formula requires the calculation of \( a, b, \text{and } c \) from \( ax^2 + bx + c \), enabling the extraction of the polynomial's remaining roots.

• Use when a cubic polynomial splits to a quadratic form.
• Helps confirm or find additional roots rapidly.
• Ensures all roots are real, verifying solutions are accurate.

It's essential for those dealing with complex solutions in polynomials.