Problem 46 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ x^{4}-81 $$

Step-by-Step Solution

Answer

The completely factored form is \((x^2 + 9)(x + 3)(x - 3)\).

Step 1: Recognize the Form

The given polynomial is a difference of squares. A difference of squares is expressed as \( a^2 - b^2 \) which factors to \( (a+b)(a-b) \). In this case, \( x^4 \) and \( 81 \) are both squares. Identify \( a = x^2 \) and \( b = 9 \).

Step 2: Apply Difference of Squares Formula

With \( a = x^2 \) and \( b = 9 \), apply the difference of squares formula: \( x^4 - 81 = (x^2 + 9)(x^2 - 9) \).

Step 3: Further Factorization

Notice that \( x^2 - 9 \) is also a difference of squares. Using the same formula as before, \( a = x \) and \( b = 3 \), apply: \( x^2 - 9 = (x + 3)(x - 3) \).

Step 4: Combine All Factors

Substitute the factorization from Step 3 into the expression from Step 2: \( x^4 - 81 = (x^2 + 9)(x + 3)(x - 3) \).

Key Concepts

Difference of Squares

The difference of squares is a special pattern in algebraic expressions that you might frequently encounter. This pattern involves the subtraction of two squared terms. The general form of a difference of squares is given as \( a^2 - b^2 \). A useful feature of this expression is that it can be factored into two binomials: \( (a + b)(a - b) \).

• The first term is \( a + b \).
• The second term is \( a - b \).

This technique simplifies expressions for further calculations. In the exercise example \( x^4 - 81 \), both terms \( x^4 \) and \( 81 \) are perfect squares. Recognizing this pattern allows you to break down complex expressions into simpler elements.

Algebraic Expressions

Algebraic expressions are fundamental components of algebra, composed of variables, constants, and operations. They can include addition, subtraction, multiplication, or division. An expression doesn't contain an equality sign; different from equations.
In the example \( x^4 - 81 \), we have an algebraic expression. Here, \( x^4 \) includes the variable \( x \) raised to the power 4, and \( 81 \) is a constant.
Manipulating algebraic expressions involves techniques like expansion, simplification, and factorization. These manipulations are crucial in solving equations or simply rewriting expressions in a different, often simpler, form. Recognizing patterns, such as the difference of squares, facilitates this manipulation, making complex operations manageable.

Factoring Techniques

Factoring is a process where an expression is rewritten as a product of its factors. It's a fundamental skill in algebra that assists in simplifying expressions and solving equations.
One common factoring technique is recognizing the difference of squares, which we've analyzed. When dealing with the expression \( x^4 - 81 \), this is applied twice.

• First, observe \( x^4 - 81 \) as a difference of squares resulting in \( (x^2 + 9)(x^2 - 9) \).
• Then, notice that \( x^2 - 9 \) itself is another difference of squares, factored into \( (x + 3)(x - 3) \).

This systematic breakdown into simpler components demonstrates the power of factoring techniques. By fully decomposing expressions into their fundamental parts, solving and understanding algebraic expressions becomes a lot easier.