Problem 56 - Algebra and Trigonometry

Question

Show that the given value(s) of \(c\) are zeros of \(P(x)\), and find all other zeros of \(P(x)\). \(P(x)=3 x^{4}-x^{3}-21 x^{2}-11 x+6, \quad c=\frac{1}{3},-2\)

Step-by-Step Solution

Answer

The zeros of \(P(x)\) are \(\frac{1}{3}, -2, \frac{-3 + \sqrt{5}}{2}, \) and \(\frac{-3 - \sqrt{5}}{2}\).

Step 1: Evaluate P(c) for c = 1/3

First, let's substitute \(c = \frac{1}{3}\) into \(P(x)\): \[P\left(\frac{1}{3}\right) = 3\left(\frac{1}{3}\right)^4 - \left(\frac{1}{3}\right)^3 - 21\left(\frac{1}{3}\right)^2 - 11\left(\frac{1}{3}\right) + 6\]. Simplifying this, we get: \[= \frac{1}{27} - \frac{1}{27} - \frac{21}{9} - \frac{11}{3} + 6\]. All terms cancel to yield \(0\), showing \(c = \frac{1}{3}\) is a zero.

Step 2: Evaluate P(c) for c = -2

Next, substitute \(c = -2\) into \(P(x)\): \[P(-2) = 3(-2)^4 - (-2)^3 - 21(-2)^2 - 11(-2) + 6\]. Simplifying, \[= 3(16) + 8 - 21(4) + 22 + 6\], \[= 48 + 8 - 84 + 22 + 6\], \[= 0\]. This confirms \(c = -2\) is also a zero.

Step 3: Factor P(x) by given zeros

Knowing \(x = \frac{1}{3}\) means a factor of \(3x - 1\) and \(x = -2\) means a factor of \(x + 2\). Therefore, \(P(x) = (3x - 1)(x + 2)Q(x)\). Use polynomial division to determine \(Q(x)\), starting by dividing \(P(x)\) by \((3x - 1)(x + 2)\).

Step 4: Perform polynomial division by (3x - 1)(x + 2)

Divide \(P(x)\) by \(3x^2 + 5x - 2\) (expanded product of known factors) to find \(Q(x)\). After completing the division, the quotient \(Q(x) = x^2 + 3x + 1\). Thus, \[P(x) = (3x - 1)(x + 2)(x^2 + 3x + 1)\].

Step 5: Find zeros of quadratic factor

Solve for \(x\) in \(x^2 + 3x + 1 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], where \(a = 1, b = 3, c = 1\). This gives: \[x = \frac{-3 \pm \sqrt{9 - 4}}{2}\], \[x = \frac{-3 \pm \sqrt{5}}{2}\]. The remaining zeros are \[x = \frac{-3 + \sqrt{5}}{2}\] and \[x = \frac{-3 - \sqrt{5}}{2}\].

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, similar to long division but applied to polynomials. This method helps simplify complex polynomials and find factors or quotient polynomials. For instance, given the polynomial \(P(x)\), and knowing certain roots like \(x = \frac{1}{3}\) and \(x = -2\) lead to factors \(3x - 1\) and \(x + 2\) respectively, polynomial division helps to find any remaining factorization.
The process involves:

• Aligning terms in decreasing powers of \(x\).
• Dividing the first term of the dividend by the first term of the divisor.
• Multiplying the entire divisor by this quotient and subtracting from the dividend.
• Repeating the steps with the new polynomial formed until all terms are covered.

This approach pinpoints any leftover factors after determining the prominent ones, aiding in breaking down the polynomial into simpler, manageable factors.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations that cannot be factored easily. It delivers the solutions or zeros of the quadratic equation of the form \(ax^2 + bx + c = 0\) and is given by:

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

In our exercise, after polynomial division, we are left with \(x^2 + 3x + 1 = 0\). Here, the quadratic formula helps us find the remaining zeros. The formula requires values for \(a, b,\) and \(c\), which correspond to the coefficients of the quadratic equation.

• The discriminant \(b^2 - 4ac\) indicates the nature of zeros. If positive, there are two real roots.
• Using \(a = 1, b = 3, c = 1\), we find the roots \(x = \frac{-3 \pm \sqrt{5}}{2}\), showcasing two distinct real zeros.

This formula is invaluable for its precision and applicability in different contexts, offering a quick pathway to identifying quadratic roots.

Polynomial Factoring

Factoring polynomials involves writing the polynomial as a product of its factors. This process simplifies the polynomial and is essential for finding its roots or zeros. In our exercise, once the zeros \(x = \frac{1}{3}\) and \(x = -2\) are verified, these provide factors \(3x - 1\) and \(x + 2\) respectively.

• The polynomial \(P(x)\) enables further reduction by utilizing identified zeros to break down the expression into simpler terms.
• After long division, we factor the quotient \(x^2 + 3x + 1\) using the quadratic formula, expanded upon in a previous section.

Ultimately, factoring gives \(P(x) = (3x - 1)(x + 2)(x^2 + 3x + 1)\). The benefits of such factorization include:

• Easy identification of polynomial zeros.
• Practicality in solving polynomial equations systematically.
• Simplifies understanding polynomial behavior near these roots.

Polynomial factoring streamlines complex expressions, offering pathways to straightforward solutions by breaking down into fundamental components.