Problem 22 - Algebra 2

Question

Find all of the zeros of each function. \(p(x)=6 x^{4}+22 x^{3}+11 x^{2}-38 x-40\)

Step-by-Step Solution

Answer

The zeros of the function are \(x = -2, x = \frac{5}{3}\), and two complex zeros, \(x = \frac{-5 \pm i\sqrt{55}}{3}\).

Step 1: Identify Potential Rational Zeros

First, apply the Rational Root Theorem to find potential rational zeros. The possible rational zeros are the factors of the constant term (-40) divided by the factors of the leading coefficient (6). Thus, the potential rational zeros are \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{20}{3}, \pm \frac{40}{3}\).

Step 2: Use Synthetic Division

Test the potential rational zeros using synthetic division to determine which, if any, are actual zeros of \(p(x)\). By testing several values, suppose \(x = -2\) is found to be a zero, meaning \(p(-2) = 0\).

Step 3: Factor the Polynomial by Synthetic Division

Perform synthetic division on \(p(x)\) using \(x = -2\) to divide the polynomial. This results in \(p(x) = (x + 2)(6x^3 + 10x^2 - 9x - 20)\).

Step 4: Find Further Zeros Using the Quotient

Now solve \(6x^3 + 10x^2 - 9x - 20 = 0\). Try additional potential zeros using synthetic division again or factor by grouping or directly using the quadratic formula for parts that can be reduced to second degree functions.

Step 5: Use the Quadratic Formula

For terms that result in quadratic polynomials, apply the quadratic formula. If \(ax^2 + bx + c = 0\), then \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 6: Check Complex Zeros

If the discriminant \((b^2 - 4ac)\) is negative, use the quadratic formula and include complex numbers \(i\) where \(i^2 = -1\).

Step 7: Verify All Zeros

Once all zeros are found, verify by substituting them back into the original polynomial to ensure that they satisfy the equation.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool to predict possible rational solutions, or "zeros," of a polynomial equation. It states that any rational zero, or root, of the polynomial equation \(a_nx^n + \, ... \, + a_1x + a_0 = 0\) must be in the form \( \frac{p}{q} \), where \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).

For instance, in the polynomial \(p(x) = 6x^4 + 22x^3 + 11x^2 - 38x - 40\), \(a_0\) is -40 and \(a_n\) is 6. This gives us potential rational zeros as \[ \pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{20}{3}, \pm \frac{40}{3} \].

• List out the factors of the constant term.
• List out the factors of the leading coefficient.
• Divide factors of the constant by factors of the leading coefficient.

By testing these possible zeros, we can determine which, if any, are actual roots of the polynomial. This theorem narrows the number of possible rational roots, saving time in calculations.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form \(x - c\). It's typically used to find zeros or factor polynomials more easily than long division. Instead of dealing with the variables, synthetic division efficiently handles just the coefficients.

How does it work?
- Write down the coefficients of the polynomial in descending order.
- Choose a potential zero \(c\), as found from using the Rational Root Theorem.
- Use synthetic division steps to simplify: bring down the leading coefficient, multiply by \(c\), and add to the next coefficient, continuing until done. If the remainder is zero, then \(c\) is indeed a zero of the polynomial.

Using the example of \(p(x) = 6x^4 + 22x^3 + 11x^2 - 38x - 40\), by testing values, like \(x = -2\), we'll see it gives a remainder of zero, confirming it's a root. This process simplifies the polynomial, giving us a reduced version to work with in further calculations.

Quadratic Formula

The quadratic formula is a catch-all tool used to find the roots of any quadratic polynomial of the form \(ax^2 + bx + c = 0\). The solution is given by the formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula is crucial when the polynomial already reduced, like after synthetic division, includes a quadratic part. The quadratic formula can find real numbers, or if the discriminant (\(b^2 - 4ac\)) is negative, it shows roots as complex numbers. This is especially useful when obstacles arise in factoring directly, providing an algebraic solution instead.

• Identify coefficients \(a\), \(b\), and \(c\).
• Calculate the discriminant \(b^2 - 4ac\).
• Plug values into the quadratic formula to find the zeros.

Combining this with the previously found zeros from methods like synthetic division ensures all possibilities are covered.

Complex Numbers

Complex numbers extend the idea of a number through the inclusion of \(i\), the imaginary unit, which satisfies \(i^2 = -1\). This becomes vital when facing a negative discriminant in the quadratic formula.

In practical application, when using the quadratic formula, a negative under the square root indicates complex roots. For example, if you encounter \(-9\) under the square root, it becomes \(3i\).

Why are complex numbers important?

• They provide a solution even if no real number solution exists.
• Essential in both pure and applied mathematics.
• They're represented as \(a + bi\), where \(a\) and \(b\) are real numbers.

Complex numbers complete the set of polynomial solutions, ensuring every polynomial has exactly as many roots as its degree. Using tools like the quadratic formula with complex numbers closes the loop on finding all zeros of a given polynomial.