Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ f(x)=x^{3}+2 x^{2}-9 x-18 $$

Step-by-Step Solution

Answer

The x-intercepts are \( x = -3, x = 3, \) and \( x = -2 \).

Step 1: Set the Function Equal to Zero

To find the x-intercepts of the polynomial function \( f(x) = x^3 + 2x^2 - 9x - 18 \), we need to set the function equal to zero and solve for \( x \):\[ x^3 + 2x^2 - 9x - 18 = 0 \]

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial is a factor of the constant term divided by a factor of the leading coefficient. The constant term here is -18, and the leading coefficient is 1. So, possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.

Step 3: Test Possible Roots

We can use synthetic division or direct substitution to test possible rational roots from Step 2. Let's first try \( x = -3 \).\( f(-3) = (-3)^3 + 2(-3)^2 - 9(-3) - 18 = -27 + 18 + 27 - 18 = 0 \).Since \( f(-3) = 0 \), \( x = -3 \) is a root.

Step 4: Perform Synthetic Division

Divide the polynomial by \( x + 3 \) using synthetic division:1. Write coefficients: 1 (from \( x^3 \)), 2 (from \( x^2 \)), -9 (from \( x \)), -18 (constant).2. Use \( -3 \) as the divisor in synthetic division: - The process yields: 1, -1, -6, 0.Thus, \[ f(x) = (x + 3)(x^2 - x - 6) \].

Step 5: Factor the Quadratic Equation

Now, factor \( x^2 - x - 6 \) completely:\( x^2 - x - 6 = (x - 3)(x + 2) \).Thus, the polynomial becomes: \( f(x) = (x + 3)(x - 3)(x + 2) \).

Step 6: Identify the x-intercepts

Set each factor equal to zero to find the x-intercepts:1. \( x + 3 = 0 \) gives \( x = -3 \).2. \( x - 3 = 0 \) gives \( x = 3 \).3. \( x + 2 = 0 \) gives \( x = -2 \).

Step 7: Conclusion: List the x-intercepts

The x-intercepts of the function \( f(x) = x^3 + 2x^2 - 9x - 18 \) are \( x = -3, x = 3, \) and \( x = -2 \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is an invaluable tool when solving polynomial equations. It provides a list of possible rational roots a polynomial might possess. This theorem hinges on the relationship between the factors of the constant term and the leading coefficient of the polynomial. In the context of our exercise, the polynomial is \( f(x) = x^3 + 2x^2 - 9x - 18 \). To apply the Rational Root Theorem, we do the following:

• Identify the constant term, which is \( -18 \).
• Identify the leading coefficient, which is \( 1 \) in our polynomial.
• The possible rational roots are then obtained by taking all the factors of \( -18 \) (i.e., \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \)) and dividing them by the factors of the leading coefficient \( 1 \).

This results in the same list of potential roots: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \). Testing these rational roots with the polynomial function can help identify actual roots, simplifying the subsequent steps of solving and factoring.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form \( x - c \), and it's particularly useful for checking potential roots from the Rational Root Theorem. Here's how it works with our polynomial \( f(x) = x^3 + 2x^2 - 9x - 18 \) and a potential root like \( x = -3 \):

• List down the coefficients of the polynomial: \( 1, 2, -9, -18 \).
• Bring down the first coefficient directly below.
• Multiply this coefficient by the root \( -3 \) and add the result to the next coefficient, continuing this process for all terms.

For \( x = -3 \), the result gives a remainder of \( 0 \), confirming it as a root. The resulting polynomial, \( x^2 - x - 6 \), is derived from the synthetic division process, preparing us for factorization in the next step of solving the polynomial.

Quadratic Factoring

After using synthetic division, our polynomial is simplified to a quadratic form: \( x^2 - x - 6 \). Quadratic factoring involves finding two binomials that multiply to form the quadratic polynomial. The steps are:

• Identify two numbers that multiply to the constant term \( -6 \) and add up to the coefficient of \( x \), which is \( -1 \).
• These numbers are \( -3 \) and \( 2 \).
• This allows us to write the quadratic as \( (x - 3)(x + 2) \).

Combining this with the known root from synthetic division, the complete factorization of the polynomial is \( (x + 3)(x - 3)(x + 2) \). Factoring not only solves the polynomial but also reveals its x-intercepts, which in this problem are \( x = -3, x = 3, \) and \( x = -2 \). These intercepts aid in understanding the polynomial's graph.