Problem 78 - Elementary and Intermediate Algebra

Question

Factor. $$ 9-9 n^{4} $$

Step-by-Step Solution

Answer

The expression \(9 - 9n^4\) factors to \(9(1-n)(1+n)(1+n^2)\).

Step 1: Recognize the Common Factor

First, identify the common factor in the expression. The expression is \(9 - 9n^4\). Notice that both terms have a common factor of 9. So, factor out 9 from the expression. This gives us: \[9(1 - n^4)\].

Step 2: Factor as a Difference of Squares

Now, observe the expression inside the parenthesis: \(1 - n^4\). This can be rewritten as a difference of squares. Recall that \((a^2 - b^2) = (a - b)(a + b)\). Here, \(1\) is \((1)^2\) and \(n^4\) is \((n^2)^2\). So, \(1 - n^4 = (1)^2 - (n^2)^2\). Factor it as: \[(1 - n^2)(1 + n^2)\].

Step 3: Simplify the Result

Substitute the factored form of \(1 - n^4\) back into the expression. We initially factored out a 9, thus our expression becomes: \[9(1 - n^2)(1 + n^2)\].

Step 4: Check for Further Factoring

Check if any of the new expressions can be factored further. The term \(1-n^2\) is again a difference of squares and can be factored as \((1-n)(1+n)\). Therefore, the fully factored form is: \[9(1-n)(1+n)(1+n^2)\].

Key Concepts

Difference of Squares

The concept of a difference of squares is a critical tool in algebra for factoring certain types of polynomial expressions. A difference of squares follows the form \(a^2 - b^2\), where both \(a\) and \(b\) are squared terms. This specific pattern allows us to factor it as \((a - b)(a + b)\).

To apply this to a real expression, take the expression \(1 - n^4\) from our problem. Here, \(1\) can be rewritten as \((1)^2\) and \(n^4\) is \((n^2)^2\). Thus, \(1 - n^4\) is essentially a difference of squares.

When you rewrite it in this form, it becomes clear that we can factor it into \((1 - n^2)(1 + n^2)\).

But don't stop there! The term \(1 - n^2\) is also a difference of squares. It can further be factored into \(1^2 - n^2 = (1 - n)(1 + n)\). This demonstrates how deep the difference of squares pattern can go in complex expressions.

Common Factoring

Common factoring is a preliminary step in simplifying expressions, where you identify a factor that is common to all terms in the expression. This process helps in reducing the expression into a simpler form that can be further factored or simplified.

In our problem, the expression given is \(9 - 9n^4\). A quick peek shows that both terms, \(9\) and \(9n^4\), share a common factor of 9. Extracting this common factor simplifies the expression to \(9(1 - n^4)\).

This technique is particularly useful as it often simplifies complex algebraic expressions, making them easier to work with. Common factoring is a foundational skill in algebra that sets the stage for further operations like the difference of squares or quadratic factoring.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra and are used in different areas of mathematics. Mastering how to manipulate them is key to solving complex problems.

In the context of our example, the expression \(9 - 9n^4\) is an algebraic expression. In this case, you are learning how to simplify it by using different factoring techniques. Through steps like common factoring and recognizing the difference of squares, you break down a complex polynomial into simpler pieces.

Understanding algebraic expressions involves recognizing patterns and applying algebraic rules, such as distributive property, commutative \(a + b = b + a\), and associative \((a + b) + c = a + (b + c)\) properties. With these skills, you're prepared to manipulate expressions and find solutions to challenging math problems.