Problem 16 - College Algebra

Question

For the following exercises, solve the following polynomial equations by grouping and factoring. $$ x^{3}+3 x^{2}-25 x-75=0 $$

Step-by-Step Solution

Answer

The roots are \(x = -3\), \(x = 5\), and \(x = -5\).

Step 1: Group the Terms

Look at the polynomial: \(x^{3}+3x^{2}-25x-75=0\). Group the terms into two pairs: \((x^3 + 3x^2)\) and \((-25x - 75)\).

Step 2: Factor Each Group

Factor out the greatest common factor from each pair. For the first group \((x^3 + 3x^2)\), the common factor is \(x^2\), which gives: \(x^2(x + 3)\). For the second group \((-25x - 75)\), the common factor is \(-25\), which gives: \(-25(x + 3)\).

Step 3: Extract the Common Binomial

Notice that both groups contain the binomial \((x + 3)\). So the expression can be factored as \((x + 3)(x^2 - 25) = 0\).

Step 4: Recognize a Difference of Squares

Observe that \(x^2 - 25\) is a difference of squares, which can be factored further as \((x - 5)(x + 5)\). The equation becomes \((x + 3)(x - 5)(x + 5) = 0\).

Step 5: Solve for the Roots

Set each factor equal to zero and solve for \(x\). Thus, \(x + 3 = 0\), \(x - 5 = 0\), and \(x + 5 = 0\) give \(x = -3\), \(x = 5\), and \(x = -5\).

Key Concepts

Factoring by Grouping

Factoring by grouping is a useful method to solve polynomial equations, especially when dealing with polynomials of higher degrees. This technique involves grouping terms with common factors to simplify the factoring process.
Here's how it works in simple steps:

• Identify pairs of terms that share a common factor. Group these terms together. In the exercise, we start with two pairs: \((x^3 + 3x^2)\) and \((-25x - 75)\).
• Factor out the greatest common factor (GCF) from each pair. A GCF is a number or term that evenly divides both terms in the group. For example, from \(x^3 + 3x^2\), the GCF is \(x^2\), resulting in \(x^2(x + 3)\).
• Repeat the process for the second group \((-25x - 75)\) to get \(-25(x + 3)\).

This simplifies the equation to reveal common binomials, ultimately leading to successful factoring and solving.

Difference of Squares

The concept of the difference of squares makes solving polynomial equations simpler. It involves expressions where two perfect squares are subtracted. The formula is given by: \ \[a^2 - b^2 = (a-b)(a+b) \]Below are key points to note:

• Perfect squares are numbers or terms like \(x^2\), 4, 9, 16, etc., which result from squaring a whole number or expression.
• The exercise provided an example of \(x^2 - 25\), where \(x^2\) and 25 are both perfect squares.
• The difference of squares formula was applied to \(x^2 - 25\), transforming it into \((x - 5)(x + 5)\).

Recognizing the difference of squares allows us to break down complex expressions, helping further solve the polynomial equation effectively.

Solving Polynomial Equations

Solving polynomial equations can appear challenging, but by using the right strategies, it becomes more manageable. The overall goal is to find the values of the variable that satisfy the equation. Here is a streamlined approach:

• Start by factoring the polynomial. Factoring methods, like grouping and using formulas for special products (e.g., difference of squares), are strategic.
• Once the equation is factored, set each factor equal to zero. This is based on the zero product property, which states if \((a \times b = 0)\), then either \(a = 0\) or \(b = 0\).
• For example, after factoring \((x + 3)(x - 5)(x + 5) = 0\), we set each factor to zero: \(x + 3 = 0\), \(x - 5 = 0\), and \(x + 5 = 0\).

Solving these simple equations determines the roots or solutions of the polynomial.

Roots of Equations

The roots of a polynomial equation are the values of \(x\) that make the equation equal to zero. Identifying these roots is crucial as they are the solutions to the equation. Let's delve into this:

• The roots are derived from setting each factor in a fully factored polynomial equation to zero and solving for \(x\).
• In the exercise, the factored equation is \((x + 3)(x - 5)(x + 5) = 0\). Solving gives roots \(x = -3\), \(x = 5\), and \(x = -5\).
• These roots represent points on the x-axis where the polynomial curve intersects the axis.

Determining the roots is a fundamental step in solving polynomial equations and understanding their behavior.