Problem 57 - 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. $$ a^{3}-24-4 a+6 a^{2} $$

Step-by-Step Solution

Answer

\((a + 6)(a + 2)(a - 2)\)

Step 1: Rearrange the Expression

Start by rewriting the expression in standard polynomial form, which is in descending order of exponents. Rearrange the given expression: \ \(a^3 - 24 - 4a + 6a^2\) to: \ \(a^3 + 6a^2 - 4a - 24\).

Step 2: Group Terms

The next step is to use grouping to try to factor the expression. Group the terms into two pairs: \ \((a^3 + 6a^2) + (-4a - 24)\).

Step 3: Factor Each Group

Factor out the greatest common factor (GCF) from each group: \ - For the first group \(a^3 + 6a^2\), the GCF is \(a^2\), so \(a^2(a + 6)\). \ - For the second group \(-4a - 24\), the GCF is \(-4\), so \(-4(a + 6)\). \ Now the expression looks like: \ \(a^2(a + 6) - 4(a + 6)\).

Step 4: Factor Out the Common Binomial Factor

Notice that \((a + 6)\) is a common factor. Factor \((a + 6)\) from the expression: \ \((a + 6)(a^2 - 4)\).

Step 5: Factor the Difference of Squares

Recognize that \(a^2 - 4\) is a difference of squares, which can be factored further: \ \(a^2 - 4 = (a + 2)(a - 2)\). \ Thus, the complete factorization of the original expression is: \ \((a + 6)(a + 2)(a - 2)\).

Key Concepts

Polynomial Expressions

Polynomials are expressions that consist of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Each term in a polynomial consists of a coefficient multiplied by a variable raised to an exponent. The expression from our original exercise, \(a^{3}-24-4a+6a^{2}\), is a polynomial as it involves powers of the variable \(a\).

When working with polynomials, it is common to rewrite them in standard form. This means arranging the terms in descending order based on the exponents. In our exercise, the polynomial \(a^3 - 24 - 4a + 6a^2\) is rearranged to \(a^3 + 6a^2 - 4a - 24\).

Polynomials can be classified by the number of terms they have, such as monomials (one term), binomials (two terms), or trinomials (three terms). This helps in recognizing patterns and applying the right techniques for factoring them more efficiently.

Understanding the structure and form of polynomial expressions is crucial as it provides a foundation for techniques like grouping and factoring, as demonstrated in the exercise above.

Greatest Common Factor

Factoring polynomials often begins with identifying the greatest common factor (GCF) in groups of terms. The GCF is the highest factor that divides all terms in a polynomial without leaving a remainder.

In our exercise, once the polynomial is rearranged, the expression is grouped as \((a^3 + 6a^2) + (-4a - 24)\). Each group is then factored separately.

• For the group \(a^3 + 6a^2\), the GCF is \(a^2\), so we factor it to get \(a^2(a + 6)\).
• For the group \(-4a - 24\), the GCF is \(-4\), giving us \(-4(a + 6)\).

This step simplifies the expression by pulling out common terms, making it easier to factor further. Recognizing the GCF is a fundamental part of simplifying polynomial expressions. It can also significantly reduce the complexity of the expressions you work with.

Difference of Squares

A difference of squares is a special polynomial form where two squares are subtracted from one another. It occurs in the format \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\).

In the final steps of our exercise, we identified a difference of squares in the expression \(a^2 - 4\). Recognizing this structure allows us to factor the term \(a^2 - 4 = (a + 2)(a - 2)\).

• This technique is a powerful tool in algebra, simplifying expressions and solving equations effectively.
• Remember that both terms must be perfect squares for this method to apply.

Understanding the concept of differences of squares provides a quick and efficient way to break down more complex polynomial expressions into simpler factors.