Problem 398 - Precalculus

Question

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ x^{3}+2 x^{2}-9 x-18=0 $$

Step-by-Step Solution

Answer

The only real zero is \(-3\).

Step 1: Write Down the Polynomial Coefficients

In the given polynomial \(x^{3}+2x^{2}-9x-18=0\), we can identify the coefficients as: \[a_3 = 1, \ a_2 = 2, \ a_1 = -9, \ a_0 = -18\]

Step 2: Identify Possible Rational Zeros

According to the Rational Zero Theorem, any potential rational zero, \(p/q\), is a ratio of the factors of the constant term \(a_0 = -18\) to the factors of the leading coefficient \(a_3 = 1\). So, possible values for \(p\) are the factors of \(-18\) which are \(\pm1, \pm2, \pm3, \pm6, \pm9, \pm18\), and \(q\) is \(\pm1\). Thus, the possible rational zeros are \(\pm1, \pm2, \pm3, \pm6, \pm9, \pm18\).

Step 3: Test Possible Zeros Using Synthetic Division

Start testing possible rational zeros using synthetic division. Begin with \(x = 1\). The synthetic division setup for \(x = 1\) would look like this:\[(1, 2, -9, -18)\]Perform the division:- Bring down the 1.- Multiply the 1 by 1 and add to 2 to get 3.- Multiply 1 by 3 and add to -9 to get -6.- Multiply 1 by -6 and add to -18 to get -24.The remainder is not zero, so \(x = 1\) is not a root.

Step 4: Continue Testing Possible Zeros

Continue testing with \(x = -1, 2, -2, 3, -3\), etc., using similar synthetic division:- Test \(x = -3\):\[(1, 2, -9, -18)\]- Bring down 1.- Multiply -3 by 1, add to 2, get -1.- Multiply -3 by -1, add to -9, get 0.- Multiply -3 by 0, add to -18, get 0.The remainder is \(0\), so \(x = -3\) is a root.

Step 5: Simplify the Polynomial

With \(x = -3\) found as a root, the polynomial can now be divided by \(x + 3\) to simplify it:Using synthetic division for the polynomial \(x^3 + 2x^2 - 9x - 18\) by \(x + 3\) gives:\[(1, 2, -9, -18)\] will simplify to:\((1(x^2) - 1(x) + 6)\)So, the simplified polynomial is \(x^2 - x + 6\).

Step 6: Solve the Quadratic Equation

Now solve the quadratic \(x^2 - x + 6 = 0\) using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute \(a = 1\), \(b = -1\), \(c = 6\):\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4\cdot1\cdot6}}{2\cdot1}\]\[x = \frac{1 \pm \sqrt{1 - 24}}{2}\]\[x = \frac{1 \pm \sqrt{-23}}{2}\]The discriminant is negative, so no further real roots exist.

Key Concepts

Polynomial Roots

Finding the roots of a polynomial is like discovering the solutions to the equation that make it equal zero. In essence, the roots are the 'answers' that satisfy the polynomial equation. For the polynomial given in the exercise, checking its roots showcases which numbers, when plugged back into the polynomial, will yield a result of zero. Using the Rational Zero Theorem is a systematic way to identify potential rational roots, which can then be tested through methods like synthetic division to confirm whether they are actual roots. Identifying roots is a crucial starting point in solving polynomial equations.

Synthetic Division

Synthetic division is a simplified form of dividing a polynomial by a linear factor of the form \(x - c\). It is particularly useful for finding roots and simplifying polynomials. In this process, you use the coefficients of the polynomial and perform arithmetic operations in a streamlined manner.

Here’s how it works:

• Write down the coefficients of the polynomial.
• Guess a potential root (in the form \(x = a\)).
• Use this potential root to perform synthetic division.
• Check if the remainder is zero. If it is, \(x = a\) is indeed a root.

This technique not only helps in confirming the roots but simplifies the polynomial for further analysis or factorization.

Quadratic Formula

Once a polynomial is simplified, the quadratic formula can help solve quadratic equations, which are of the form \(ax^2 + bx + c = 0\).

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula provides the roots of any quadratic equation and can discern the nature of these roots based on the discriminant \(b^2 - 4ac\). A negative discriminant, as found in the solution, indicates that the polynomial has no further real roots, but might have complex roots. Thus, the quadratic formula becomes an essential tool in solving polynomial equations after factorization or simplification.

Complex Numbers

When a polynomial has a negative discriminant, it implies that there are no real solutions, and the roots are complex numbers. Complex numbers consist of a real part and an imaginary part, formulated as \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\).

In this problem, the quadratic formula yielded a negative discriminant, leading to complex roots for the equation \(x^2 - x + 6 = 0\). These complex roots are expressed as:

• \(x = \frac{1 + \sqrt{-23}}{2}\)
• \(x = \frac{1 - \sqrt{-23}}{2}\)

Understanding complex numbers helps in comprehensively solving more advanced polynomials, which often appear in various fields of mathematics and engineering. They allow for the extension of real numbers to solve equations that do not have real solutions.