Problem 97 - College Algebra

Question

Solve each equation. $$x(x+1)^{3}-42(x+1)^{2}-0$$

Step-by-Step Solution

Answer

The practical solution to the equation is \(x = 0\). The other solution would come from the cubic equation, but it can't be solved analytically with standard algebraic techniques, requiring numerical methods instead.

Step 1: Simplify the Equation

First, simplify the given equation \(x(x+1)^{3}-42(x+1)^{2}=0\) by expanding the terms inside the brackets. This gives us \(x^{4}+3x^{3}+3x^{2}+x-42x^{2}-42x=0\).

Step 2: Combine Like Terms

Next, combine the like terms in the equation. Upon combining like terms, we obtain the equation \(x^{4}+3x^{3}-x^{2}-41x=0\).

Step 3: Factorizing

As we see, all terms in the equation have \(x\) as common factor, then we can simplify the equation further by factorizing with respect to \(x\). Factorizing we get \(x(x^{3}+3x^{2}-x-41)=0\).

Step 4: Apply Zero-product Property

To find the solution for \(x\), we use the zero-product property. According to the zero-product property, if the product of two numbers is zero, then at least one of the numbers must be zero. Hence we set each factor equal to zero and solve for \(x\). Now we set each factor equal to zero to obtain the solutions. This gives us \(x = 0\) and \(x^{3}+3x^{2}-x-41 = 0\). Unfortunately, the cubic equation cannot be solved analytically and requires numerical methods. Hence, \(x = 0\) is the only practical solution in this case.

Key Concepts

Zero-Product Property

The Zero-Product Property is a crucial concept when solving polynomial equations, especially when dealing with factored expressions. This property states that if a product of two or more factors is zero, then at least one of the factors must be zero. It sounds simple, but it simplifies solving polynomial equations significantly.

When you encounter a factored polynomial equation such as \( x(x+1) = 0 \), you can use the Zero-Product Property to find the roots easily. Here’s how it works:

• Set each factor in the equation equal to zero.
• Solve each resulting equation to find the solutions of the equation.

This method is particularly handy because it allows us to break down complex equations into simpler, more manageable parts. For example, when you have an equation \( ab = 0 \), either \( a = 0 \) or \( b = 0 \) (or both). This principle helps in zeroing in on solutions efficiently.

Factoring Polynomial Expressions

Factoring polynomial expressions involves breaking down complex polynomial terms into simpler, multiplied forms that are easier to manage and solve. This process is essential for solving complex equations that would otherwise be difficult to tackle.

Here are the steps to factor a polynomial:

• Identify common factors in the terms of the polynomial.
• Extract the common factor and rewrite each term accordingly to simplify the polynomial.
• Continue factoring until no further factors can be extracted.

For instance, with the polynomial \( x^4 + 3x^3 - x^2 - 41x \), notice that all terms contain \( x \) as a factor. By factoring out \( x \), the expression simplifies to \( x(x^3 + 3x^2 - x - 41) \). This step is crucial, as it sets the stage for applying the Zero-Product Property to find the solutions to the equation.

Effective factoring not only simplifies the equation but is also the stepping-stone for further solving techniques like quadratic and cubic solutions in polynomials.

Cubic Equations

Cubic equations are polynomial equations of degree three, typically expressed in the form \( ax^3 + bx^2 + cx + d = 0 \). These equations generally require more complex techniques to solve than quadratic equations.

While some cubic equations can be factored easily into simpler polynomial expressions, others may need numerical methods or special formulas for a solution. The example given in the original exercise demonstrates this: beyond factoring out the common term \( x \), you're left with \( x^3 + 3x^2 - x - 41 = 0 \), which is more challenging to solve analytically.

• Common approaches to solve cubic equations involve the use of the Factor Theorem, Rational Root Theorem, or specific formulae designed for cubics.
• In many cases, if an algebraic solution is intractable, numerical methods such as the Newton-Raphson method can approximate the roots successfully.
• Recognize when a cubic equation cannot be easily simplified further, and employ numerical or graphing methods to find practical solutions.

Understanding and identifying when a cubic equation can be simplified or when numerical methods are needed is a key skill in algebra.