Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ f(x)=x^{3}-18 x $$

Step-by-Step Solution

Answer

The roots are \( x = 0 \), \( x = 3\sqrt{2} \), and \( x = -3\sqrt{2} \).

Step 1: Identify Common Factors

First, identify any common factors in the polynomial. In the function \( f(x) = x^3 - 18x \), both terms have a common factor of \( x \). So, we can factor out \( x \) from the polynomial: \( f(x) = x(x^2 - 18) \).

Step 2: Solve for Root from Factored Term

The factor \( x \) in \( x(x^2 - 18) \) indicates that one of the roots is \( x = 0 \).

Step 3: Factor the Quadratic

Next, address the quadratic expression \( x^2 - 18 = 0 \). This can be solved either by factoring or using the quadratic formula. In this case, notice that \( x^2 = 18 \) can be rewritten as \( x = \pm \sqrt{18} \).

Step 4: Simplify the Square Roots

Simplify \( \sqrt{18} \) further. Since \( 18 = 9 \times 2 \), we have \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \). Hence the roots for \( x^2 - 18 = 0 \) are \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \).

Step 5: List All Roots

Finally, list all the roots found from the function \( f(x) = x^3 - 18x \). The roots of the function are \( x = 0 \), \( x = 3\sqrt{2} \), and \( x = -3\sqrt{2} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a math problem into smaller, more manageable parts. Think of it as finding the ingredients that multiply together to make a final product. The goal is to express the polynomial as a product of simpler polynomials or factors.
For our problem, starting with the polynomial function \( f(x) = x^3 - 18x \), you'll first look for common factors. Here, both terms share the factor \( x \).
Once you take this factor out, the polynomial simplifies to \( f(x) = x(x^2 - 18) \). This process of breaking down helps identify quicker ways to solve equations.

• Identify common factors in the polynomial.
• Express the polynomial as a product of these factors.

By factoring, you're making it easier to find the roots, or solutions, of the polynomial equation.

Quadratic Formula

The quadratic formula is a nifty tool for solving quadratic equations that take the form \( ax^2 + bx + c = 0 \). It is especially handy when factoring isn't straightforward. The formula is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]This allows you to find the roots (solutions) of any quadratic equation with a few straightforward substitutions.
In our case, the quadratic \( x^2 - 18 = 0 \) is already set for use with the quadratic formula. For simplicity, let's consider:
- \( a = 1 \)- \( b = 0 \)- \( c = -18 \)
Plug these into the formula to find your solutions. However, you might notice here that you don't need it since the equation simplifies directly by isolating \( x^2 \).
The beauty of the quadratic formula is its reliability—it works even when other methods are cumbersome.

Simplifying Square Roots

Simplifying square roots involves breaking them down into their simplest form. This makes equations easier to work with and solutions quicker to evaluate. When you have a square root, you're looking for whole numbers or simpler terms that can multiply under the root to give you the original number.
Consider the square root \( \sqrt{18} \). You can simplify it by finding numbers whose square is a factor of 18. Notice that 18 is \( 9 \times 2 \), and \( 9 \) is a perfect square.

• Write \( \sqrt{18} \) as \( \sqrt{9 \times 2} \).
• Since \( \sqrt{9} = 3 \), it simplifies to \( 3\sqrt{2} \).

Through simplification, you get the simplest form of a root, which is often neater and more understandable in the context of solving equations. In our scenario, this helps express the roots as \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \), providing neat and accurate roots of the quadratic.