Problem 24 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ a b^{2}-4 a+3 b^{2}-12 $$

Step-by-Step Solution

Answer

The expression is factored as \((a + 3)(b - 2)(b + 2)\).

Step 1: Group the Terms

First, rearrange the expression in a way that allows us to see grouping possibilities. The expression is: \[ ab^2 - 4a + 3b^2 - 12 \] Rearrange it to:\[ (ab^2 + 3b^2) + (-4a - 12) \]

Step 2: Factor by Grouping

In the grouped expression \((ab^2 + 3b^2) + (-4a - 12)\), factor out the common factor in each group:- From \(ab^2 + 3b^2\), factor out \(b^2\): \[ b^2(a + 3) \]- From \(-4a - 12\), factor out \(-4\): \[ -4(a + 3) \]

Step 3: Identify the Common Factor

Now, notice that both groups have a common factor of \((a + 3)\). So you can factor \((a + 3)\) out of the entire expression:\[ (a + 3)(b^2 - 4) \]

Step 4: Fully Factor the Expression

Check if \((b^2 - 4)\) can be factored further. This is a difference of squares, which can be written as:\[ (b - 2)(b + 2) \]Thus, the fully factored form of the expression is:\[ (a + 3)(b - 2)(b + 2) \]

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that involves numbers, variables, and operations. Unlike an equation, it doesn't have an equal sign but can include multiple terms connected by addition or subtraction.

• **Terms:** Different parts of an expression separated by plus or minus signs. Each term can have coefficients (numbers), variables (letters), and exponents (powers).
• **Coefficients:** Numbers that multiply the variables in a term. In the term \(3b^2\), 3 is the coefficient.
• **Variables:** Symbols used to represent unknown values, like \(a\) and \(b\) in the given expression.
• **Exponents:** Indicate how many times a number or variable is multiplied by itself, such as \(b^2\).

Understanding how these components work together is crucial before starting to factor the expression.
It allows us to recognize how terms can be grouped and factored efficiently.

Factoring Techniques

Factoring is a process used to break down expressions into products of simpler expressions or numbers. For complex algebraic expressions, several common techniques help simplify the process:

• **Factoring by Grouping:** Involves rearranging and grouping terms to find common factors. This method is useful when expressions have no obvious single common factor, as seen in the example expression \((ab^2 + 3b^2) + (-4a - 12)\).
• **Finding Common Factors:** Look for the greatest common factor in terms within the group to simplify. In the grouped terms \(ab^2 + 3b^2\), \(b^2\) is factored out, and in \(-4a - 12\), \(-4\) is factored out.

By utilizing these techniques, any complex looking expression can often be transformed into a simpler, factored form. This simplification aids in further solving and understanding of algebraic problems.

Difference of Squares

The difference of squares is a specific form of factoring for expressions that can be written as \(x^2 - y^2\). This expression is factored into the product \((x - y)(x + y)\), and it is often used after other methods of factoring have been applied.

• **Identifying Difference of Squares:** Look for two square terms separated by a minus sign. This is a common structure in algebraic expressions that can often be overlooked.
• **Applying the Formula:** The difference of squares factored results in two binomials. For example, \(b^2 - 4\) was broken down to \((b - 2)(b + 2)\).

Recognizing the difference of squares in an expression allows us to quickly and efficiently simplify it further.
This method is especially important when fully factoring expressions, ensuring no further simplification is missed.