Problem 70 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 3 x^{3}+5 x^{2}-6 x-10 $$

Step-by-Step Solution

Answer

The expression factors to \((3x + 5)(x^2 - 2)\).

Step 1: Group Terms

First, we will group the terms into two pairs in order to factor by grouping. The expression is \(3x^3 + 5x^2 - 6x - 10\). Let's group \((3x^3 + 5x^2)\) and \((-6x - 10)\).

Step 2: Factor Each Group

Factor out the greatest common factor (GCF) from each group. For the first group \((3x^3 + 5x^2)\), the GCF is \(x^2\). Factoring it out, we get \(x^2(3x + 5)\). For the second group \((-6x - 10)\), the GCF is \(-2\). Factoring it out, we get \(-2(3x + 5)\). The expression becomes \(x^2(3x + 5) - 2(3x + 5)\).

Step 3: Factor Out the Common Binomial Term

Notice that \((3x + 5)\) is a common factor in both terms after grouping. Factor \((3x + 5)\) out of the expression. This gives us the factored expression: \((3x + 5)(x^2 - 2)\).

Step 4: Verify the Factorization

Expand \((3x + 5)(x^2 - 2)\) to confirm that it equals the original expression. Expanding, we have: \((3x + 5)(x^2) + (3x + 5)(-2) = 3x^3 + 5x^2 - 6x - 10\). Since this matches the original expression, the factorization is verified as correct.

Key Concepts

Polynomial

A polynomial is a mathematic expression consisting of variables (also known as indeterminates), coefficients, and exponents arranged in terms according to certain rules. Each polynomial is composed of one or more terms, where each term includes:

• A coefficient, which is a numeric constant,
• A variable (such as \( x \)), and
• An exponent, which indicates the power to which the variable is raised.

For example, in the polynomial \( 3x^3 + 5x^2 - 6x - 10 \), there are four terms:

• \( 3x^3 \) (a cubic term),
• \( 5x^2 \) (a quadratic term),
• \( -6x \) (a linear term), and
• \( -10 \) (a constant term).

Polynomials can be classified based on the number of terms they contain:

• A monomial, which has only one term,
• A binomial, with two terms, and
• A trinomial, which consists of three terms.

Polynomials are fundamental in algebra because they allow for the construction of equations and expressions, helping in various mathematical operations, like addition, subtraction, and especially factorization.

Greatest Common Factor

The greatest common factor (GCF) in algebra refers to the largest factor that can evenly divide each term in a set of terms. Identifying the GCF is a crucial step in the process of polynomial factorization.

In our exercise, the polynomial \( 3x^3 + 5x^2 - 6x - 10 \) is divided into groups such as \( (3x^3 + 5x^2) \) and \( (-6x - 10) \). Within each of these groups, we look for the GCF:

• For the first group, \( (3x^3 + 5x^2) \), the GCF is \( x^2 \), since both terms contain at least two \( x's \) and no greater numerical factor.
• For the second group, \( (-6x - 10) \), the GCF is \( -2 \), which is the highest number that evenly divides both terms.

Once the GCF is identified, it can be factored out of each group. This step simplifies subsequent operations, enabling other methods like binomial factorization to work effectively.

Binomial Factorization

Binomial factorization involves rewriting a polynomial expression as the product of two binomials. A binomial is simply a polynomial with two terms, such as \( x + y \).

In our example, after factoring the GCF from each group, the expression is transformed into \( x^2(3x + 5) - 2(3x + 5) \). Here, \((3x + 5)\) emerges as a common binomial factor in both terms after grouping. This allows us to factor them into a single expression:

• \((3x + 5)(x^2 - 2)\).

This process is critical because it reveals underlying simpler expressions.

To verify binomial factorization, one needs to expand \((3x + 5)(x^2 - 2)\). Upon expansion, you should retrieve the original polynomial, affirming the correctness of the factorization process.

This makes binomial factorization an invaluable technique, especially in simplifying complex algebraic expressions, solving equations, and graphing polynomial functions.