Problem 15 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$49 y^{4}-16$$

Step-by-Step Solution

Answer

The difference of squares \(49y^{4} - 16\) factors to \((7y^2 + 4)(7y^2 - 4)\).

Step 1: Identify the squares

Look at both terms of the equation \(49y^{4} - 16\). The first term, \(49y^{4}\), is the square of \(7y^2\), and the second term, \(16\), is the square of \(4\). Therefore, \(a = 7y^2\) and \(b = 4\).

Step 2: Apply the difference of squares rule

Plug the values of \(a\) and \(b\) from Step 1 into the formula for the difference of squares. This will give \((7y^2 + 4)(7y^2 - 4)\).

Step 3: Check your work

To ensure that you've factored the expression correctly, you can expand your result by using the FOIL (First, Outer, Inner, Last) method. This should give you back the original expression \(49y^{4} - 16\).

Key Concepts

Polynomial Factoring

Polynomial factoring is like breaking down a complex mathematical sentence into its basic words or phrases. Just as a sentence is composed of words, a polynomial is composed of terms, and the goal is to rewrite the polynomial as a product of simpler expressions. For example, factoring the difference of squares involves recognizing that certain polynomials are the result of multiplying two 'conjugate' expressions, which are binomials with the same terms but opposite operations—like \( (x + y)(x - y) \). The expression \(49y^{4} - 16\) fits this pattern, as it can be seen as the subtraction of two squared terms.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and are made up of constants, variables, and algebraic operations (addition, subtraction, multiplication, and division). In the context of factoring, an expression like \(49y^{4} - 16\) can be thought of as a miniature mathematical puzzle. Figuring out this puzzle involves understanding the relationships between the terms of the expression, which can be key to simplifying the equation and solving algebraic problems.

FOIL Method

The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, and Last, describing the order in which you multiply the terms of the binomials. For instance, to confirm that \( (7y^2 + 4)(7y^2 - 4) \) is indeed an accurate factoring of \(49y^{4} - 16\), multiply the 'First' terms \(7y^2\) and \(7y^2\), the 'Outer' terms \(7y^2\) and \( -4\), the 'Inner' terms \(4\) and \(7y^2\), and finally, the 'Last' terms \(4\) and \( -4\). The method helps to ensure that the factored form expands back to the original, confirming its correctness.

Square of a Term

The square of a term simply means multiplying the term by itself. For example, squaring \(7y^2\) gives \(49y^4\), just as squaring \(4\) gives \(16\). Recognizing a term's square in algebraic expressions, such as in the difference of squares \(49y^{4} - 16\), is crucial. It's the first step in factoring, as it reveals the underlying structure that can be used to simplify the expression using specific factoring formulas.