Problem 58 - Algebra and Trigonometry

Question

\(57-58\) . 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

Zeros of P(x) are \(\frac{1}{3}, -2, \frac{1}{2}, \approx -2.167\).

Step 1: Verify if c=1/3 is a Zero of P(x)

Substitute \(c = \frac{1}{3}\) into the polynomial \(P(x)\). Compute \(P\left(\frac{1}{3}\right) = 3\left(\frac{1}{3}\right)^4 - \left(\frac{1}{3}\right)^3 - 21\left(\frac{1}{3}\right)^2 - 11\left(\frac{1}{3}\right) + 6\). Evaluate the expression. The result is zero, confirming that \(c = \frac{1}{3}\) is a zero of \(P(x)\).

Step 2: Verify if c=-2 is a Zero of P(x)

Similarly, replace \(x\) with \(-2\) in \(P(x)\). Compute \(P(-2) = 3(-2)^4 - (-2)^3 - 21(-2)^2 - 11(-2) + 6\). Evaluate this expression, and it should also equal zero, proving that \(c = -2\) is a zero of \(P(x)\).

Step 3: Perform Polynomial Division

With known zeros \(x = \frac{1}{3}\) and \(x = -2\), we factor \(P(x)\) by dividing it by \(3x-1\) and \(x + 2\). Use synthetic division or long division to simplify \(P(x)\) step-by-step, confirming the divisions result in zero remainders each time.

Step 4: Factor the Remainder Polynomial

After successful division, the remainder polynomial is \(3x^2 + ax + b\). Solve any quadratic equation that remains using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find any remaining zeros.

Step 5: List All Zeros of P(x)

Combine all found zeros: the confirmed zeros \(\frac{1}{3}\) and \(-2\), and the zeros found from solving the quadratic. List them as the complete solution to finding all zeros of \(P(x)\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial of lesser degree. It helps us analyze and simplify the polynomial for further factorization and zero identification. There are two primary techniques to perform polynomial division: long division and synthetic division. Both methods aim to reduce the polynomial into a simpler form, breaking it down into manageable parts.

• **Long Division**: Similar to numerical long division, this method involves dividing the terms step-by-step, subtracting and iterating through the polynomial.
• **Synthetic Division**: A more streamlined process, especially for dividing by linear factors. It is often quicker and involves less writing compared to long division.

Polynomial division sets the stage for further solving by transforming complex problems into easier ones. It is particularly useful when you want to find additional zeros of a polynomial, as it reduces the degree of the polynomial and can be performed repeatedly until a quadratic (or simpler) polynomial is obtained.

Synthetic Division

Synthetic division is a shortcut method used to divide a polynomial by a linear factor of the form \((x - c)\). This technique simplifies many of the steps present in long division and is convenient for polynomials with integer coefficients.
To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial you want to divide.
• Identify the zero of the divisor, i.e., if you are dividing by \((x - c)\), use \(c\) in synthetic division.
• Bring down the first coefficient.
• Multiply this coefficient by \(c\), and write the product under the next coefficient. Add this product to that coefficient.
• Continue this pattern until all coefficients have been processed.
• The final row of numbers contains the coefficients of the quotient, and potentially, the remainder.

Synthetic division is particularly useful when checking if a given value is a zero of the polynomial because it makes it straightforward to see whether the remainder is zero. When conducting synthetic division with our example, you'd see quickly that the zeroes \(\frac{1}{3}\) and \(-2\) lead to no remainder, verifying these as true zeros.

Quadratic Formula

The quadratic formula is a powerful tool for finding the zeros of a quadratic polynomial, i.e., a polynomial of the form \(ax^2 + bx + c\). When division reduces a polynomial to quadratic form, this formula becomes incredibly handy.
The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, the components of the quadratic equation are plugged in, and the formula outputs the possible values of \(x\) that make the equation equal to zero.

• The **discriminant** \(b^2 - 4ac\) determines the nature of the roots:
  • If it's positive, there are two distinct real roots.
  • If it's zero, there is one real root (a repeated root).
  • If it's negative, there are two complex roots.

In our example, once polynomial division reduces the polynomial down to a quadratic, using the quadratic formula helps find the remaining roots, completing the set of zeros for \(P(x)\).

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into a product of simpler polynomials. This method can be incredibly useful for finding the zeros of the polynomial and further understanding its characteristics.

• **Simple Factoring**: When there are obvious common factors between terms, these can be taken out to simplify the polynomial.
• **Factoring by Grouping**: A technique often used when the polynomial has four terms, grouping terms to combine them into factored expressions.
• **Special Formulas**: Recognizing patterns like the difference of squares \((a^2-b^2 = (a-b)(a+b))\) or perfect square trinomials \((a^2 \pm 2ab + b^2 = (a \pm b)^2)\).

Factoring ensures that once a polynomial is simplified through polynomial division, we'd further break it down to linear factors, making zero-finding trivial. In our quadratic reduced form, applying these methods can efficiently give the last set of zeros.