Problem 76 - Algebra and Trigonometry

Question

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 the Terms

Group the terms of the polynomial in pairs to facilitate factoring by grouping. The expression is \( 3x^3 + 5x^2 - 6x - 10 \). Group it as \((3x^3 + 5x^2) + (-6x - 10)\).

Step 2: Factor out the Greatest Common Factor from Each Group

From the first group \(3x^3 + 5x^2\), factor out \(x^2\), resulting in \(x^2(3x + 5)\). From the second group \(-6x - 10\), factor out \(-2\), resulting in \(-2(3x + 5)\).

Step 3: Factor out the Common Binomial Factor

Notice that both groups contain the common binomial factor \((3x + 5)\). Factor this out to get \((3x + 5)(x^2 - 2)\).

Step 4: Check for Further Factorization

Check if the expression \((x^2 - 2)\) can be factored further. It cannot be factored further using real numbers as it is a difference of squares involving \(x^2\) and a prime number \(2\). The completely factored form is \((3x + 5)(x^2 - 2)\).

Key Concepts

Factoring by Grouping

Factoring by grouping is a powerful technique used to factor polynomials, especially when the polynomial has four terms. This method involves grouping terms in pairs and then factoring out the greatest common factor from each pair. It simplifies expressions by reducing them into products of simpler binomial factors.

To start, group the polynomial terms into two pairs. For instance, take the polynomial expression:

• \(3x^3 + 5x^2 - 6x - 10\)

Group it into

• \((3x^3 + 5x^2) + (-6x - 10)\)

This strategic grouping sets the stage for factoring out a common factor from each pair. It helps reveal a common binomial factor that can be factored out in the subsequent steps. Once we have grouped terms correctly, the expressions inside each group will often share common elements, making them easier to factor further.

Greatest Common Factor

The greatest common factor, or GCF, is the largest factor that divides each term in an expression without leaving a remainder. Calculating the GCF allows us to simplify expressions efficiently and is fundamental when factoring by grouping.

In our example, with the expression split into two groups:

• \(3x^3 + 5x^2\)
• \(-6x - 10\)

For the first group, \(3x^3 + 5x^2\), the GCF is \(x^2\), as both terms contain \(x^2\) as a factor. Factoring \(x^2\) out gives us:

• \(x^2(3x + 5)\)

For the second group, \(-6x - 10\), the GCF is \(-2\). Factoring out \(-2\) results in:

• \(-2(3x + 5)\)

By finding and factoring out these GCFs from each pair, we discover a common binomial factor \((3x + 5)\), allowing us to rewrite the entire expression as a product of two simpler factors.

Difference of Squares

The concept of the difference of squares is a special factoring case involving expressions of the form \(a^2 - b^2\), which equals \((a + b)(a - b)\). This identity is useful for certain quadratic expressions but not applicable to all polynomials.

In our problem, after factoring by grouping and simplifying, we are left with the expression:

• \((3x + 5)(x^2 - 2)\)

At first glance, \(x^2 - 2\) might resemble a difference of squares. However, the number 2 is not a perfect square, so further factoring using real numbers is not possible. It requires recognizing when an expression can be reduced using the difference of squares and when it cannot, as in the case of \(x^2 - 2\), which remains in its simplest form within real numbers after attempting to apply known factoring methods.