Problem 56 - Elementary and Intermediate Algebra

Question

Factor. $$ -121 b^{4}+36 a^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(-(6a - 11b^2)(6a + 11b^2)\).

Step 1: Recognize the form

Observe that the expression \(-121b^4 + 36a^2\) resembles a difference of squares, which follows the pattern: \(x^2 - y^2 = (x+y)(x-y)\). Here, consider how each term can be rewritten as a square.

Step 2: Express each term as a square

Rewrite \(-121b^4\) as \(-(11b^2)^2\) and \(36a^2\) as \((6a)^2\). Hence, the given expression becomes \(-(11b^2)^2 + (6a)^2\).

Step 3: Apply difference of squares

Apply the difference of squares formula to \(-(11b^2)^2 + (6a)^2\).This gives: \((6a + 11b^2)(6a - 11b^2)\).

Step 4: Address the negative sign factor

Factor out \(-1\) from \(-(11b^2)^2\). We have:\(-1((11b^2)^2 - (6a)^2)\)This doesn't change Step 3 factoring but clarifies the negative sign source.

Key Concepts

Difference of Squares

In algebra, the "Difference of Squares" is a specific factoring technique that applies to expressions that take the form \(x^2 - y^2\). This pattern is unique because it can always be broken apart into two binomials:

• \((x + y)(x - y)\)

This formula is especially helpful because it simplifies the process of breaking down more complex polynomial expressions.

To identify a difference of squares in a given expression, you should look out for:

• Two perfect square terms (meaning each can be expressed as something squared)
• A subtraction sign between them

Once this form is recognized, applying the factoring formula becomes straightforward, making complex algebraic expressions much easier to work with.

Factoring Techniques

Factoring techniques are strategies used to rewrite algebraic expressions in simpler or more useful forms. These techniques are essential in solving algebra problems and understanding how expressions relate to each other. There are several common methods used in factoring:

• Greatest Common Factor (GCF): This involves finding the largest factor that two or more terms have in common and dividing each term by this factor.
• Difference of Squares: Particularly useful for expressions formed like \(x^2 - y^2\), as mentioned previously.
• Quadratic Trinomials: For expressions like \(ax^2 + bx + c\), techniques such as factoring by grouping or using the quadratic formula may apply.

Factoring is used not only to simplify expressions but also to make equations easier to solve. Recognizing the right approach, such as identifying a difference of squares, can streamline problem-solving.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler "factor" polynomials whose product gives the original polynomial. This is a crucial skill in algebra, allowing you to solve equations, analyze functions, and simplify expressions.

Key steps in polynomial factoring include:

• Identifying Patterns: Look for recognizable forms such as the difference of squares or perfect squares.
• Simplifying Terms: Express each term in its simplest square form especially when dealing with squares and cubes.
• Applying Standard Formulas: Use knowledge of algebraic identities to split the polynomial into factors.

Understanding polynomial factoring, and knowing when to apply techniques like the difference of squares makes simplifying and calculating large expressions much more manageable. It requires practice to become familiar with identifying and applying the right factoring method for different expressions.