Problem 34 - Algebra A Combined Function

Question

Factor each completely. $$ 49 y^{2}+9 $$

Step-by-Step Solution

Answer

The expression cannot be factored over the reals.

Step 1: Identify the Expression Type

The expression is a sum of squares: \(49y^2 + 9\). This can be rewritten as \((7y)^2 + 3^2\). The sum of squares does not factor further using real numbers.

Step 2: Confirm if Special Factoring is Possible

Check if the expression fits any special factoring formulas, like differences of squares or perfect square trinomials. In this case, it doesn't match any such patterns that allow for factoring over the real numbers.

Step 3: Conclusion

There is no further factoring possible for the expression \(49y^2 + 9\) over the set of real numbers. The expression is a sum of squares, which is irreducible in the realm of real numbers.

Key Concepts

Factoring Techniques

Factoring is a fundamental technique used in algebra to simplify expressions or solve equations. It involves expressing a polynomial as a product of smaller polynomials. Understanding different factoring techniques can help you tackle seemingly complex expressions with ease.

To factor an expression completely, one must often try different methods, such as:

• Factoring out the Greatest Common Factor (GCF): This involves identifying and dividing out the highest common factor that all terms share.
• Difference of Squares: If you have an expression of the form \(a^2 - b^2\), it factors into \((a + b)(a - b)\).
• Perfect Square Trinomial: An expression like \(a^2 + 2ab + b^2\) can be factored as \((a + b)^2\).
• Sum or Difference of Cubes: These special forms can also be factored using specific formulas.

Each factoring method is like a tool in your mathematical toolkit, and choosing the right one depends on the expression's form.

Expression Types

Expressions in algebra can take various forms, and identifying the type of expression is crucial in determining whether it can be factored. Some common expression types include:

• Monomials: Single-term expressions, e.g., \(5x\).
• Binomials: Two-term expressions, e.g., \(x^2 - 4\).
• Trinomials: Three-term expressions, e.g., \(x^2 + 5x + 6\).
• Polynomials: Expressions with multiple terms, such as \(x^3 - 2x + 1\).

In the exercise, we identified \(49y^2 + 9\) as a *sum of squares*. Unlike the *difference of squares*, where factors are readily available, the sum of squares does not factor over the reals. Recognizing expression types helps in applying the appropriate factoring technique.

Irreducible Polynomials

When a polynomial cannot be factored further over a certain number set (like the real numbers), it is called an irreducible polynomial. For example, \(49y^2 + 9\) is identified as a sum of squares.

This particular form remains unfactorable because there is no simple algebraic expression using real numbers that can make it a product of other polynomials. In other cases, polynomials are irreducible due to higher complexities, or because factorization requires numbers from another set, like complex numbers. Understanding irreducible polynomials is essential, as it informs you when you're likely dealing with an expression that doesn't factor in your current number system. As such, comprehending this helps in focusing efforts on appropriate mathematical strategies, saving time and frustration.