Problem 56 - Algebra 2

Question

Solve each equation. $$ 4 x^{3}-16 x^{2}+12 x=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(4x^{3}-16x^{2}+12x=0\) are \(x=0\), \(x=1\), and \(x=3\).

Step 1: Factor Out the Greatest Common Factor (GCF)

First, identify the greatest common factor of the terms in the equation. In this case, it is clear that each term has an 'x' term and can also be divided by 4. Thus, we factor out the GCF, which is \(4x\), from each term in the equation.

Step 2: Rewrite the Equation After Factoring

After factoring out the \(4x\), the equation becomes \(4x(x^2 - 4x + 3) = 0\).

Step 3: Apply the Zero-Product Property

Since the equation is of the form \(a \times b = 0\), we can apply the Zero-Product Property which states that if the product of two factors is zero, then at least one of the factors must be zero.

Step 4: Set Each Factor Equal to Zero

Setting each factor equal to zero gives us two separate equations: \(4x = 0\) and \(x^2 - 4x + 3 = 0\).

Step 5: Solve for x from the First Factor

Solving \(4x = 0\) yields \(x = 0\).

Step 6: Solve the Quadratic Equation

The quadratic equation \(x^2 - 4x + 3 = 0\) can be factored into \((x-1)(x-3) = 0\). Setting each of these factors to zero gives us two additional solutions: \(x = 1\) and \(x = 3\).

Step 7: Combine All Solutions

Combine the solutions from the first factor and the quadratic equation to get the complete set of solutions for the original equation.

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra, useful in simplifying expressions and solving equations. When a polynomial is factored, it is written as the product of two or more polynomials.

Take the polynomial equation from the exercise, \(4x^3 - 16x^2 + 12x = 0\). The first step is to identify common factors in each term. Recognizing these and factoring them out simplifies the polynomial, making it easier to solve. For instance, each term here contains an \(x\) and can be divided by 4, making \(4x\) the greatest common factor.

Zero-Product Property

The zero-product property is invaluable when solving polynomial equations. This principle states that if the product of two factors equals zero, then at least one of the factors must be zero.

After factoring our original equation, we have \(4x(x^2 - 4x + 3) = 0\). According to the zero-product property, we can set each factor equal to zero and solve for \(x\). This step leads to \(4x = 0\) and \(x^2 - 4x + 3 = 0\), creating simpler equations to find the values of \(x\) that satisfy the original equation.

Quadratic Equations

Quadratic equations, often written in the form \(ax^2 + bx + c = 0\), are second-degree polynomials and can have either two, one, or no real number solutions.

In the step-by-step solution, the equation \(x^2 - 4x + 3 = 0\) is a quadratic equation. It can be factored into \(x-1\) and \(x-3\), using techniques such as trial and error or the FOIL method for factoring. Solving these factors for zero gives the possible values for \(x\).

Greatest Common Factor

The greatest common factor (GCF) refers to the largest number or expression that divides into all terms of a polynomial without leaving a remainder. Identifying the GCF is often the first step in factoring and simplifying polynomials.

In our exercise, the GCF of \(4x^3 - 16x^2 + 12x\) is \(4x\). By factoring out the GCF, the equation is reduced and can then be separated into simpler components that are easier to solve, paving the way to finding the roots of the polynomial.