Problem 73 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$1-16 x^{4}$$

Step-by-Step Solution

Answer

The polynomial \(1 - 16x^4\) factors completely as \((1 - 2x)(1 + 2x)(1 + 4x^2)\), with \(1 + 4x^2\) not factorable with integers.

Step 1: Recognize Difference of Squares

Notice that the expression \(1 - 16x^4\) is a difference of squares. It can be rewritten as \((1)^2 - (4x^2)^2\). This means we can apply the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a = 1\) and \(b = 4x^2\) into the difference of squares formula: \((1)^2 - (4x^2)^2 = (1 - 4x^2)(1 + 4x^2)\).

Step 3: Factor Each Resulting Binomial

The first term, \(1 - 4x^2\), is itself a difference of squares. Apply the difference of squares formula again: \(1 - (2x)^2\) can be factored as \((1 - 2x)(1 + 2x)\). The second term, \(1 + 4x^2\), cannot be factored further using integers or real numbers because it does not represent the difference or sum of squares that allows further factorization.

Step 4: Write the Completely Factored Form

Combine the factored expressions to get the fully factored form of the polynomial: \((1 - 2x)(1 + 2x)(1 + 4x^2)\).

Key Concepts

Difference of Squares

When it comes to mastering polynomial factorization, one of the key techniques you will encounter is the "difference of squares." This concept is quite straightforward. Whenever you have an expression of the form \(a^2 - b^2\), it can be factored into \((a-b)(a+b)\). This is powerful because it allows us to break down more complex polynomials into simpler binomials.

In the example from the exercise, notice that \(1 - 16x^4\) is a difference of squares. You can rewrite it as \((1)^2 - (4x^2)^2\). By recognizing this pattern, you simplify the expression to \((1 - 4x^2)(1 + 4x^2)\). The beauty of this method is that it often works with perfect squares, making difficult polynomials much easier to handle. Be sure to look for squares in any polynomial to see if this technique can be applied.

Polynomial Factorization Techniques

Polynomial factorization is a fundamental skill in algebra. It involves breaking down a polynomial into simpler parts, or factors, that when multiplied together, will give you the original polynomial. There are several techniques to factor polynomials, each handy for different kinds of expressions.

• Factoring out the greatest common factor (GCF)
• The difference of squares
• The sum and difference of cubes
• Grouping

In our example, we focused on the difference of squares. But remember, no two polynomials are the same. You might need to combine several techniques to effectively factor other expressions. Recognizing which particular technique to use often requires experience and practice. Always start by checking if there's a common factor in all terms or if the expression is a special case like the difference of squares. With practice, factoring will become an easier and quicker process.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. These expressions are the building blocks of algebra, and understanding them is crucial for solving equations and grasping more complex math concepts.

In algebra, you often need to manipulate and simplify these expressions to find a solution. This is where factoring comes into play. Factoring turns a complicated expression into a product of simpler expressions. In our example, \(1 - 16x^4\) involves both numbers and a variable \(x\). When we factor it, we change the expression from potentially complicated to a product of binomials: \((1 - 2x)(1 + 2x)(1 + 4x^2)\). Each of these binomials represents a different factor of the original expression, making it easier to solve equations or find values that satisfy it. Understanding how to work with and manipulate algebraic expressions is key to success in algebra and beyond.