Problem 14 - Elementary Algebra

Question

Factor by grouping. $$ x 3-6 x 2+x-6 $$

Step-by-Step Solution

Answer

The factored form is \((x - 6)(x^2 + 1)\).

Step 1: Group Terms

The given polynomial is \( x^3 - 6x^2 + x - 6 \). First, split this polynomial into two groups: \( (x^3 - 6x^2) \) and \( (x - 6) \). This is done to prepare for factoring.

Step 2: Factor Out Common Factors From Each Group

Factor out the greatest common factor from each group separately. From the first group \( x^3 - 6x^2 \), factor out \( x^2 \), resulting in \( x^2(x - 6) \). From the second group \( x - 6 \), factor out 1 to keep it unchanged, resulting in \( 1(x - 6) \).

Step 3: Combine the Factored Groups

Now that both groups have a common factor of \( (x - 6) \), combine them: \( x^2(x - 6) + 1(x - 6) \). This shows that \( (x - 6) \) is a common factor.

Step 4: Factor Out the Common Binomial

Since \( (x - 6) \) is the common factor, factor it out: \( (x - 6)(x^2 + 1) \).

Step 5: Confirm Factoring is Complete

Check the factored expression \( (x - 6)(x^2 + 1) \). Ensure that it simplifies to the original polynomial when expanded. This confirms the factorization is complete.

Key Concepts

Factor by Grouping

When dealing with polynomials that have several terms, factor by grouping can be a useful strategy to simplify the expression. It involves organizing the polynomial into smaller groups or pairs that can be factored separately. This method often helps when the polynomial doesn't have a common factor across all terms, but does within certain groups.

Here's a brief guide on how to effectively factor by grouping:

• Break down the polynomial into two or more groups of terms that you suspect might share a common factor.
• Factor out the greatest common factor (GCF) from each group. This brings out any common terms within each group.
• If successful, you will often have a binomial or another kind of expression in common between the groups, which can then itself be factored out.

Factor by grouping is particularly advantageous in expressions where simple methods do not easily reveal the factors. It requires a bit of creativity and practice to master, but it can simplify polynomials that seem complex at first glance.

Polynomials

Polynomials are expressions made up of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. They come in various forms, such as monomials, binomials, trinomials, and beyond, depending on how many terms are present.

The structure of a polynomial is key to analyzing and simplifying it. An example of a basic polynomial is:

• Monomial: \( 5x^2 \)
• Binomial: \( x^2 - 4 \)
• Trinomial: \( x^2 + 3x + 2 \)

Understanding the degree of the polynomial is important. The degree is the highest power of the variable in the polynomial, which often determines the polynomial's behavior and the solutions possible for equations set equal to zero. In the example from our original task, \( x^3 - 6x^2 + x - 6 \) is a polynomial of degree 3.

Polynomials can be factored, expanded, and simplified to uncover more about the variable solutions involved. They are foundational in algebra, providing insight not only in equations but in various higher applications like calculus.

Common Factors

The concept of common factors is essential in algebra, particularly when simplifying expressions or solving equations. Common factors refer to terms or factors that are shared between two or more numbers or expressions.

Here’s how you can determine and use common factors in polynomials:

• Examine the terms in each group of the polynomial to find factors they share. This can include numbers and variables.
• Factoring involves pulling out these common factors to see if the rest of the polynomial can be simplified further.
• In our example exercise, the expression \( x^3 - 6x^2 + x - 6 \) utilized factoring by grouping to find that \( x - 6 \) was a common factor in both groups.

Finding common factors can simplify polynomials considerably, leading to a factored form that’s easier to handle. It's a small step with potentially big mathematical benefits, especially when further solving equations or performing operations on polynomial expressions.