Problem 21 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 4 a^{2}-64 $$

Step-by-Step Solution

Answer

Answer: The factored form of the binomial \(4a^2 - 64\) is \(4(a - 4)(a + 4)\).

Step 1: Identify common factors

In the given expression, \(4a^2 - 64\), both terms have a common factor of 4. We can factor out the 4 from both terms to simplify the expression.

Step 2: Factor out the common factor

Factoring out the common factor (4) from the given expression, we get: $$ 4(a^2 - 16) $$

Step 3: Identify potential patterns

Now, we have \(4(a^2 - 16)\), and we can observe that the expression inside the parentheses is a difference of squares. The difference of squares is a pattern for binomials of the form \(a^2 - b^2\) and can be factored as \((a + b)(a - b)\). In our expression, \(a^2 = a^2\) and \(b^2 = 16 = (4)^2\), so \(a = a\) and \(b = 4\).

Step 4: Factor the difference of squares

Using the difference of squares pattern, we can factor the expression inside the parentheses, which is \(a^2 - 16\). Thus: $$(a - 4)(a + 4)$$

Step 5: Combine the factors

Now that we have factored the expression, we can combine all the factors: $$ 4(a - 4)(a + 4) $$ So the factored form of the given binomial, \(4a^2 - 64\), is \(4(a - 4)(a + 4)\).

Key Concepts

Difference of Squares

The term 'difference of squares' refers to an algebraic expression of the form \(a^2 - b^2\). This type of expression is special because it can be factored into two binomials, \((a + b)(a - b)\). This pattern comes from multiplying two conjugates, which cancel out the middle terms, leaving only the squares of \(a\) and \(b\) with opposite signs.

For example, in our exercise, the expression inside the parentheses is \(a^2 - 16\). We recognize it as a difference of squares where \(a = a\) and \(b = 4\) because \(16 = 4^2\). So, we factor it as \((a - 4)(a + 4)\).

Remember:

• Look for two terms separated by a minus sign.
• Both terms should be perfect squares.
• Apply the formula \((a^2 - b^2) = (a + b)(a - b)\).

This method simplifies complex expressions by breaking them into simpler linear factors.

Common Factor

Factoring out a common factor is often the first step in simplifying polynomial expressions. A common factor is a number or variable that divides all terms in the expression evenly.

In our exercise, the given expression is \(4a^2 - 64\). We started by identifying that each term shares a factor of 4. By factoring out this common factor, the expression becomes simpler: \(4(a^2 - 16)\). This process not only simplifies the expression but also reveals any hidden patterns, such as the difference of squares in our expression.

Key points to remember:

• Always check for a greatest common factor (GCF) in each term.
• Factoring out the GCF should always be your initial step if possible.
• Once factored out, recheck for additional factoring patterns, like the difference of squares.

Factoring out common factors makes complex polynomial functions more manageable.

Binomials

Binomials are algebraic expressions containing exactly two terms, often joined by a plus or minus sign. They are the building blocks of more complex polynomial expressions.

In our exercise, we worked with the binomial \(4a^2 - 64\). After simplifying by factoring out the common factor, we further factored \(a^2 - 16\) into \((a - 4)(a + 4)\), showing us two resulting binomials.

Working effectively with binomials involves understanding the various patterns that can simplify their factorization:

• Recognize special patterns like difference of squares.
• Understand how to apply the distributive property in reverse to factor expressions.
• Know that the resulting binomials are expressions of the form \( (x + n) \, \text{or}\, (x - n) \).

Mastering these concepts can help in effectively tackling exercises that involve polynomial expressions and their factorization.