Problem 78 - Introductory Algebra for College Students

Question

Factor by grouping. $$x y+6 x+2 y+12$$

Step-by-Step Solution

Answer

The factored form of the given expression is \((y+6)(x+2)\).

Step 1: Rearrange the Expression

Observe the expression \(x y+6 x+2 y+12\). Rearrange the terms to prepare for grouping. The expression becomes: \(x y+6 x+2 y+12\).

Step 2: Group the Terms

Group terms that have common factors. The groups are: \((x y+6 x)\) and \((2 y+12)\).

Step 3: Factor out the Common Factors

Factor out the common factors in each group. The expression becomes: \(x(y+6) + 2(y+6)\).

Step 4: Final Factoring

Now, observe that the resulting expression has a common factor of \((y+6)\). Factor this out to get the final factored expression: \((y+6)(x+2)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of letters called variables, numbers known as coefficients, and operation symbols. For example, in the expression xy + 6x + 2y + 12, x and y are the variables, numbers like 6 and 2 are coefficients, and the addition sign + is the operation symbol. These expressions are the heart of algebra and form the basis for equations and functions.

Common Factors

The term 'common factors' refers to numbers or expressions that divide exactly into two or more other numbers or terms. When you are looking to factor by grouping, like with the expression xy + 6x + 2y + 12, you identify elements within each term that can be divided without a remainder. In the given exercise, x is a common factor in the terms xy and 6x, and the number 2 is a common factor in 2y and 12.

Factorization Techniques

Factorization is the process of breaking down a complex expression into simpler parts that, when multiplied together, give back the original expression. Techniques include finding common factors, using special formulas like the difference of squares, or factoring by grouping. In our example, factoring by grouping involves rearranging the expression and then extracting the common factors from each group. This leads to a simpler expression such as x(y + 6) + 2(y + 6) and eventually to the fully factored form (y + 6)(x + 2), using the distributive property.

Polynomial Factoring

Polynomial factoring applies to algebraic expressions like the one given in the exercise. A polynomial has multiple terms, and factoring it requires that you break it down into products of factors that are themselves less complex polynomials or monomials. Factoring by grouping is a versatile technique for polynomials with four or more terms, as it involves creating pairs of terms that share a common factor, then factoring those out. Once grouped and factored separately, if each group shares a common binomial factor, it can be factored out completely, simplifying the entire polynomial significantly, such as in our final factored form (y + 6)(x + 2).