Problem 75 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ k^{3}+k^{2}-20 k=0 $$

Step-by-Step Solution

Answer

The solutions are \( k = 0, -5, 4 \).

Step 1: Factor the Equation

The given equation is \( k^3 + k^2 - 20k = 0 \). The first step is to factor out the common term \( k \). This gives us: \[ k(k^2 + k - 20) = 0 \].

Step 2: Solve for the First Factor

Since we have factored the equation, we can solve for \( k = 0 \) from the first factor \( k \). This is one solution of the equation.

Step 3: Factor the Quadratic Equation

The remaining factor is \( k^2 + k - 20 = 0 \). To factor this quadratic, we need to find two numbers that multiply to \(-20\) and add to \(1\). These numbers are \(5\) and \(-4\), so the equation factors to \((k + 5)(k - 4) = 0\).

Step 4: Solve the Factored Quadratic

Set each factor equal to zero: \( k + 5 = 0 \) yields \( k = -5 \) and \( k - 4 = 0 \) yields \( k = 4 \).

Step 5: Collect All Solutions

The solutions to the original equation are \( k = 0, k = -5, \) and \( k = 4 \).

Key Concepts

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of factoring, we aim to express this equation as a product of two binomials. This is key because it allows us to find the solutions, or roots, of the equation easily.

To factor a simple quadratic equation like \( x^2 + x - 20 = 0 \), we look for two numbers that multiply to the constant term \( c = -20 \) and add to the linear coefficient \( b = 1 \). The process involves:

• Identifying pairs of numbers that multiply to \(-20\).
• Testing each pair to see which one adds up to \( 1 \).

The correct pair in this case is \( 5 \) and \(-4\), leading to the factorization \( (x + 5)(x - 4) = 0 \). By setting each factor equal to zero, \( x + 5 = 0 \) and \( x - 4 = 0 \), you can solve for \( x \). This method simplifies our pathway to the solutions \( x = -5 \) and \( x = 4 \).

Using this approach is foundational for understanding more complex equations and enhances problem-solving skills by breaking down complex expressions into simpler, solvable parts.

Cubic Equation

Cubic equations involve polynomial expressions in the form \( ax^3 + bx^2 + cx + d = 0 \), where the highest power of the variable \( x \) is three. These equations can be more complex than quadratics but can still be tackled by employing factoring strategies.

The exercise provided an opportunity to understand the initial step in solving a cubic equation by factoring out the greatest common factor (GCF). In this case, the term \( k^3 + k^2 - 20k = 0 \) has a GCF of \( k \). This common factor is factored out to simplify the equation to \( k(k^2 + k - 20) = 0 \).

Factoring out the GCF reduces the complexity of the equation by transforming it into a simpler form, often a quadratic, which can then be addressed using additional factoring techniques or the quadratic formula if necessary. It effectively breaks the problem into smaller, more manageable parts, making the overall solution process much more straightforward.

Roots of Polynomial

Finding the roots, or solutions, of a polynomial is a fundamental concept in algebra. The roots are the values of the variable that make the polynomial equal to zero. For any polynomial, determining these roots involves factoring the polynomial expression and solving the resulting equations.

For the given polynomial \( k^3 + k^2 - 20k = 0 \), the roots of the equation are found by first factoring the expression as a product of simpler terms: \( k(k^2 + k - 20) = 0 \). Solving \( k = 0 \) gives one root directly. The quadratic \( k^2 + k - 20 = 0 \) is further factored into \( (k + 5)(k - 4) = 0 \), yielding two more roots when solving each individual equation: \( k = -5 \) and \( k = 4 \).

In summary, the roots reflect the x-values where the polynomial intercepts the x-axis. These intercepts are significant for graphing the polynomial and understanding its behavior. Mastering finding roots equips students with tools to analyze and interpret the characteristics of polynomial functions fully.