Problem 16 - Brief Applied Calculus

Question

Solve each equation by factoring. [Hint for Exer cises 19-22: First factor out a fractional power.] $$ 3 x^{4}+12 x^{2}=12 x^{3} $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = 2 \).

Step 1: Move all terms to one side

The given equation is \( 3x^4 + 12x^2 = 12x^3 \). Start by moving all terms to one side of the equation to set it equal to zero: \( 3x^4 + 12x^2 - 12x^3 = 0 \).

Step 2: Factor out the greatest common factor

Look for the greatest common factor (GCF) in the polynomial. The GCF of \( 3x^4, 12x^2, \) and \(-12x^3\) is \( 3x^2 \). Factor \( 3x^2 \) out of the equation: \( 3x^2(x^2 + 4 - 4x) = 0 \).

Step 3: Rearrange and factor the quadratic

Rearrange the quadratic within the parentheses to standard form: \( x^2 - 4x + 4 \). Recognize this as a perfect square trinomial and factor it as \( (x-2)^2 \), so the equation becomes \( 3x^2(x-2)^2 = 0 \).

Step 4: Solve for the roots

Apply the Zero Product Property which states that if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \). Here, solve \( 3x^2 = 0 \) and \( (x-2)^2 = 0 \). Solving \( 3x^2 = 0 \) gives \( x = 0 \). Solving \( (x-2)^2 = 0 \) gives \( x = 2 \).

Step 5: Conclude the solutions

The solutions to the original equation are \( x = 0 \) and \( x = 2 \). Verify by substituting them back into the original equation to ensure they satisfy the equation.

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables, coefficients, and exponents arranged in a particular form. They can have one or more terms, all of which are added, subtracted, or multiplied together. Each term in a polynomial includes:

• A coefficient, which is a number that multiplies the variable.
• A variable, which is often represented as 'x' or any other letter.
• An exponent, which shows the power to which the variable is raised.

For example, in the term \(3x^4\), '3' is the coefficient, 'x' is the variable, and '4' is the exponent. Understanding the structure of polynomials is crucial when dealing with algebraic expressions and equations, as they form the foundation for further operations such as factoring and solving equations.

Quadratic Equations

Quadratic equations are a type of polynomial equation that can be written in the standard form \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants, and 'a' is not zero. The characteristic feature of a quadratic equation is the squared term \(x^2\).
These equations often represent parabolas when graphed on a coordinate plane, and they have various methods of solution, including:

• Factoring, which involves expressing the quadratic as a product of two binomials.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square, a method used to make the quadratic expression a perfect square trinomial.

The goal in solving quadratic equations is to find the values of 'x' that satisfy the equation, which are known as the roots or solutions.

Zero Product Property

The Zero Product Property is a fundamental principle that simplifies solving equations involving multiple terms. It states that if the product of two or more factors equals zero, then at least one of the factors must be zero. In mathematical terms, if \(ab = 0\), then either \(a = 0\) or \(b = 0\) or both.
This property is particularly useful when solving factored equations. For instance, if an equation can be expressed in the form \((x-2)(x+3) = 0\), then, according to the Zero Product Property, \(x-2 = 0\) or \(x+3 = 0\), leading to solutions \(x = 2\) or \(x = -3\).
This concept is integral to solving quadratic equations after factoring, as it allows for finding solutions by setting each factor to zero and solving the resulting simpler equations.

Common Factor Extraction

Common factor extraction is a technique used to simplify equations or expressions by identifying and factoring out the greatest common factor (GCF). The GCF is the highest number and/or highest power of any variables that can evenly divide all terms in an expression.
For example, in the expression \(3x^4 + 12x^2 - 12x^3\), the GCF is \(3x^2\). Factoring out \(3x^2\) results in \(3x^2(x^2 + 4 - 4x)\). By extracting the common factor, you reduce the complexity of the expression, making it easier to further manipulate, solve, or factor.
This method is a crucial step in solving polynomial equations, particularly when aiming to apply the Zero Product Property. Factoring out the GCF sets the stage for further actions like factoring of quadratics or other polynomials, thereby simplifying the path to finding solutions.