Problem 14 - College Algebra

Question

Factor by grouping. $$ x^{3}+6 x^{2}-2 x-12 $$

Step-by-Step Solution

Answer

The factored form of the given expression \(x^{3}+6 x^{2}-2 x-12\) by grouping is \((x + 6)(x^{2} - 2)\)

Step 1: Grouping

First, group the terms in pairs: \((x^{3} + 6x^{2}) + (-2x - 12)\)

Step 2: Factor Out the Common Factor

For each pair, factor out the greatest common factor. In the first pair, the greatest common factor is \(x^{2}\), yielding \(x^{2}(x + 6)\). In the second pair, the greatest common factor is -2, yielding -2(x + 6). This results in the expression: \(x^{2}(x + 6) - 2(x + 6)\)

Step 3: Factor by Grouping

Factor out the common binomial term \((x + 6)\) to get the factored form of the polynomial: \((x + 6)(x^{2} - 2)\)

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components or 'factors' that, when multiplied together, give back the original polynomial. This is similar to finding the prime factors of integers, except that the factors here are algebraic expressions. The method used depends on the polynomial, but common approaches include factoring out the greatest common factor, factoring by grouping, difference of squares, sum and difference of cubes, and using the quadratic formula for quadratic equations. An essential skill for solving equations and simplifying expressions,
factoring also helps in understanding polynomial functions' behavior.

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the highest factor that divides two or more non-zero polynomials without leaving a remainder. In algebra, identifying the GCF can greatly simplify the process of reducing fractions or factoring polynomial expressions. For instance, when factoring polynomials, we look for the highest degree of each variable and the highest coefficient number that is common to all terms in an expression. By 'pulling out' this GCF, we can often make further factoring steps, like grouping, more manageable and transparent.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example is the quartic polynomial \( x^{4} + 5x^{3} - 2x^{2} + 7x - 8 \). Understanding the structure of these polynomials—how they are put together and how they can be broken down into factors—is central to many areas of algebra, including solving equations and analyzing functions. Each term of a polynomial can include a constant, a variable, or multiple variables multiplied together, which together form an orderly set of 'building blocks' that create the polynomial.

Algebraic Techniques

Algebraic techniques are the various methods used to manipulate and solve algebraic equations and expressions. They can be as simple as adding like terms or as complex as using the quadratic formula to solve a polynomial equation. Other techniques include completing the square, using the distributive property, and applying the laws of exponents. A well-grounded understanding of these techniques allows for the solving of a wide range of mathematical problems. In fact, factoring by grouping, as in the given example, is one such technique that simplifies polynomials with four or more terms into more easily solvable forms by grouping and factoring them.