Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ f(x)=2 x^{3}+2 x^{2}-4 x $$

Step-by-Step Solution

Answer

The roots of the function are \( x = 0, -2, \) and \( 1 \).

Step 1: Factor Out Common Factor

The function is given as \( f(x) = 2x^3 + 2x^2 - 4x \). First, we notice that each term in the function has a common factor of \( 2x \). Factoring \( 2x \) out, we get: \[ f(x) = 2x(x^2 + x - 2) \] This expression provides a simpler polynomial to solve for roots.

Step 2: Factor the Quadratic Expression

Next, we need to factor the quadratic expression \( x^2 + x - 2 \). To factor this, we look for two numbers that multiply to \(-2\) (the constant term) and add to \(1\) (the linear coefficient). These numbers are \(2\) and \(-1\). Thus, we factor the quadratic as follows: \[ x^2 + x - 2 = (x + 2)(x - 1) \]

Step 3: Set Each Factor Equal to Zero

Now that the function is factored completely as \( f(x) = 2x(x + 2)(x - 1) \), we set each factor equal to zero to find the roots of the function:1. \( 2x = 0 \) 2. \( x + 2 = 0 \) 3. \( x - 1 = 0 \)

Step 4: Solve for Roots

Solve each equation from Step 3 for \( x \):1. \( 2x = 0 \) yields \( x = 0 \).2. \( x + 2 = 0 \) yields \( x = -2 \).3. \( x - 1 = 0 \) yields \( x = 1 \).Therefore, the roots of the function are \( x = 0, -2, \) and \( 1 \).

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool in algebra for finding the roots of a quadratic equation. A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
When factoring is difficult or impossible, the quadratic formula becomes invaluable. The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]By using this formula, we can easily find the roots of the quadratic equation by substituting the values of \(a\), \(b\), and \(c\). Let's break it down:

• \(b^2 - 4ac\) is called the discriminant.
• The discriminant tells us the nature of the roots (real and distinct, real and same, or complex).
• The symbol \(\pm\) gives us the two roots of the equation.
  • If the discriminant is positive, there are two distinct real roots.
  • If it is zero, there is one real root (a repeated root).
  • If it is negative, the roots are complex.

Understanding the quadratic formula makes it easier to solve any quadratic equation effectively. It becomes second-hand when you practice applying this repeatedly.

Solving Cubic Equations

Solving cubic equations can be more complex than solving quadratics, but there are still several methods to find their roots. A cubic equation is usually in the form \(ax^3 + bx^2 + cx + d = 0\).
For some cubic equations, like the one in the original exercise, factoring is often a good starting point. To factor a cubic equation step-by-step:

• Look for any common factors in all terms. This makes the equation simpler.
• Write the equation as a product of factors. This can reduce a complex equation into simpler, smaller degree polynomials.
• Apply the zero product property by setting each factor to zero, which solves for its roots.

If factoring is not feasible, other methods like synthetic division or Cardano's method may be used. These techniques can handle more challenging cubic equations when direct factoring is not possible.
Be mindful of the possibility of multiple real roots or complex roots in cubic equations. It's always good to check your factorization by multiplying the factors to confirm that you end up with the original polynomial. This verification step ensures no mistakes were made during the factoring process.

Algebraic Expressions

Algebraic expressions are essential building blocks in algebra. These expressions may consist of variables, constants, and operations. Here's what to look out for:

• Variables: Typically represented by letters (e.g., \(x, y\)), they are the unknowns we often solve for.
• Coefficients: Numbers that multiply variables. For instance, in \(3x\), 3 is the coefficient.
• Constant terms: Numbers on their own (e.g., \(5\) in \(2x + 5\)).

Operations like addition, subtraction, multiplication, and division are applied to variables and constants in these expressions. Simplifying these combined elements is critical to solving algebraic problems. When dealing with algebraic expressions, it's important to be familiar with:

• Simplifying expressions by combining like terms.
• Using distributive property to expand or factor expressions.
• Rearranging expressions to make equations easier to solve.

Understanding how different terms interact in an expression — such as polynomial terms multiplying together or canceling each other out — strengthens the foundation for recognizing patterns and predicting outcomes in algebraic equations. This foundational knowledge facilitates more advanced algebraic manipulations.