Problem 53 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Answer: The fully factored form of the expression \(a^4 - b^4\) is \((a^2 + b^2)(a + b)(a - b)\).

Step 1: Identify the difference of squares

Examine the given expression: $$ a^{4} - b^{4} $$ Notice that it is a difference of squares, with the two squares being \(a^4\) and \(b^4\).

Step 2: Apply the difference of squares formula

Recall the difference of squares formula, which is given by: $$ A^2-B^2=(A+B)(A-B) $$ Comparing to our expression, we can consider \(A = a^2\) and \(B = b^2\). Applying the formula to the given expression, we get: $$ a^{4} - b^{4} = (a^2 + b^2)(a^2 - b^2) $$

Step 3: Check if factors can be factored further

Now, let's take a look at the factors obtained in Step 2: \((a^2 + b^2)\) and \((a^2 - b^2)\). - The first factor, \((a^2 + b^2)\), cannot be factored any further using real numbers as it is a sum of squares. - The second factor, \((a^2 - b^2)\), is again a difference of squares, so we can apply the difference of squares formula once more. Let \(A = a\) and \(B = b\) this time. So we get: $$ a^2 - b^2 = (a + b)(a - b) $$

Step 4: Write the final factored expression

Putting together the factored expressions from Step 2 and Step 3, we get the final factored form of the given expression as: $$ a^4 - b^4 = (a^2 + b^2)(a + b)(a - b) $$

Key Concepts

Difference of Squares

The concept of the difference of squares is a vital tool in factorization. It involves expressions where one term is subtracted from another, both of which are perfect squares. The classic formula used is:

• \( A^2 - B^2 = (A + B)(A - B) \).

This formula states that the difference between two squares can be factored into a product of a sum and a difference. The initial appearance of such an expression might look complicated, but recognizing it as a difference of squares can simplify the process significantly. With practice, spotting these expressions can become intuitive.

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients. They are composed of terms combined using addition, subtraction, and sometimes multiplication. For instance, the expression \( a^4 - b^4 \) has the following characteristics:

• Two terms: \( a^4 \) and \( b^4 \).
• Exponents represent the degree of each term.
• Terms can sometimes be combined or factored depending on their form.

In general, recognizing the structure of a polynomial helps in deciding which approach to use for factoring or simplifying it. Many polynomials, like the given example, follow special patterns such as squares or cubes that allow for streamlined factorization.

Binomials

A binomial is a type of polynomial that contains exactly two terms. In the exercise example \( a^4 - b^4 \), it is clear that we deal with a binomial:

• The two terms are \( a^4 \) and \( -b^4 \).
• These terms are connected by either addition or subtraction. Here, subtraction forms a binomial difference.

Binomials are often straightforward to factor when they match specific patterns, such as the difference of squares. Identifying it as a binomial enables the application of methods like the one discussed in the exercise to factor it effectively.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's a fundamental aspect of understanding and solving polynomials:

• Allows the representation of general relationships using expressions and equations.
• In this exercise, algebraic manipulations enable us to factor complex expressions like \( a^4 - b^4 \).
• Understanding algebraic principles is crucial for breaking down expressions into simpler components.

Factorization, such as applying the difference of squares, showcases the power of algebra in simplifying and solving problems that might appear complex at first glance. Mastering algebraic techniques allows for efficient problem-solving and deeper mathematical understanding.