Problem 16 - College Algebra

Question

In Exercises \(11-16,\) factor by grouping. $$x^{3}-x^{2}-5 x+5$$

Step-by-Step Solution

Answer

The factored form of the given polynomial is \((x^{2}-5)(x-1)\)

Step 1: Group Terms

First, divide the polynomial \(x^{3}-x^{2}-5 x+5\) into two groups so it looks like this: \((x^{3}-x^{2})+(-5 x+5)\)

Step 2: Factor Out Common Factor

Then, find the greatest common factor in each group and factor it out: \(x^{2}(x-1)-5(x-1)\)

Step 3: Factor Out the Common Binomial

Finally, look for the common binomial and factor it out, resulting in: \((x^{2}-5)(x-1)\)

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions that involve sums and differences of terms, which are formed by combining variables and coefficients using mathematical operations like addition, subtraction, and multiplication. Each term in a polynomial is made up of a coefficient (a number) multiplied by a variable raised to an exponent (a power). The general form of a polynomial can be written as:

• \( a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{2}x^{2} + a_{1}x + a_{0} \)

In this expression, \(a_{n}, a_{n-1}, ... , a_{0}\) are coefficients and \(n\) is a non-negative integer called the degree of the polynomial. Each term's exponent determines its degree, and the term with the highest exponent signifies the polynomial's overall degree.

For example, in the polynomial given in the exercise, \(x^{3} - x^{2} - 5x + 5\), the degree is 3 because the highest exponent is 3. Understanding the structure of polynomials is essential when using methods like factoring by grouping. It helps recognize patterns and possibilities for breaking down expressions into more manageable parts.

Finding Common Factors

Finding common factors is a key step in simplifying mathematical expressions, especially when dealing with polynomials. A common factor refers to a term or number that can evenly divide each term in an expression without leaving a remainder. When factoring, identifying these common factors helps simplify the expressions by dividing out these shared components.

In solving polynomials, we often focus on identifying the greatest common factor (GCF), which is the largest factor shared by all terms. Once identified, this factor can be "factored out" from the expression.

• For instance, consider the groups in the polynomial presented, namely \((x^{3}-x^{2})\) and \((-5 x+5)\).
• In \(x^{3}-x^{2}\), the terms share \(x^{2}\) as a common factor.
• In \(-5x + 5\), both terms share the number \(-5\).

When these common factors are factored out, the expression becomes easier to work with, setting the stage for further factoring techniques like binomial factoring.

The Principle of Binomial Factoring

Binomial factoring is a method employed to simplify expressions further, especially once common factors have been identified. It involves finding and factoring common binomial terms, which are expressions with two terms, across different parts of a polynomial.

After initial grouping and factoring in the solution, we are left with an expression containing two binomial terms that look similar. In the exercise at hand, the groups \(x^2(x-1)\) and \(-5(x-1)\) reveal that \((x-1)\) is a repeating factor. This common binomial can then be factored out. This results in:

• \((x^2 - 5)(x-1)\)

This step capitalizes on recognizing patterns within the polynomial structure, allowing the expression to be factored further neat and cleanly.

Binomial factoring reduces complexity and is a strategic approach in solving higher-degree polynomials. It harnesses the power of simplification by focusing on these common patterns, making it much more straightforward to handle polynomial equations.