Problem 28 - College Algebra

Question

For the following exercises, factor the polynomial. $$ 324 x^{2}-121 $$

Step-by-Step Solution

Answer

The polynomial is factored as \((18x + 11)(18x - 11)\).

Step 1: Identify the structure

Examine the polynomial to determine if it has the structure of a difference of squares. The polynomial is \( 324x^2 - 121 \). Notice that it consists of two terms separated by a minus sign.

Step 2: Write as a difference of squares

Recognize that \( 324x^2 \) is \((18x)^2\) and \(121\) is \((11)^2\). The polynomial can be expressed as a difference of squares: \((18x)^2 - (11)^2\).

Step 3: Apply the difference of squares formula

Use the formula for factoring a difference of squares: \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = 18x\) and \(b = 11\).

Step 4: Write the factored form

Substitute \(a\) and \(b\) into the difference of squares formula to get the factored form: \((18x + 11)(18x - 11)\).

Key Concepts

Difference of Squares

Difference of squares is a special technique in algebra that simplifies the factoring of certain types of polynomials. When you see a polynomial with two terms separated by a minus sign, like in our example, it's important to check if it can be expressed as a difference of squares.

A difference of squares takes the form of \( a^2 - b^2 \), where both \( a \) and \( b \) are squares of some expressions. The beauty of this method lies in its simplicity.

• The formula for factoring a difference of squares is: \( a^2 - b^2 = (a + b)(a - b) \).
• It always results in two binomial factors.
• Each factor is composed of the square roots of the terms for \( a^2 \) and \( b^2 \).

In our exercise, the polynomial \( 324x^2 - 121 \) fits perfectly into this format because \( 324x^2 \) is \( (18x)^2 \) and \(121\) is \( (11)^2 \). By recognizing it as a difference of squares, we can easily factor it using the mentioned formula.

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves breaking down a polynomial into simpler terms, or factors, that when multiplied together give the original polynomial. This process can help solve equations, simplify expressions, and find roots.

When dealing with polynomials, there are different methods to factor them. For a difference of squares, the method is straightforward, but there are other techniques for different forms:

• **Greatest Common Factor (GCF):** Start by checking if there's a common factor in all the terms of the polynomial. Factor it out first.
• **Grouping:** For polynomials with four or more terms, try to group them into pairs or sets that can each be factored further.
• **Trinomials:** For polynomials in the form \( ax^2 + bx + c \), you may use the quadratic formula or factor by trial and error to find factors that multiply to \( ac \) and add to \( b \).

The context defines which method would be most effective. Identifying special patterns like the difference of squares simplifies the process considerably, as seen in the provided exercise.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. These expressions can be as simple as \( 2x + 3 \) or as complex as the polynomial in our example, \( 324x^2 - 121 \). Understanding algebraic expressions is foundational to making sense of mathematical problems and solutions.

Key concepts in understanding algebraic expressions include:

• **Terms:** Parts of the expression separated by addition or subtraction operators.
• **Coefficients:** The numerical part multiplied by the variable. In \( 324x^2 \), the coefficient is 324.
• **Variables:** Symbols used to represent unknowns, typically \( x, y, z \), etc.
• **Exponents:** These indicate how many times a variable is multiplied by itself. For example, in \( x^2 \), the exponent is 2.

Mastering the language of algebraic expressions allows for analysis and manipulation of these expressions, leading to successful factoring and problem-solving. Recognizing different forms and patterns, such as the difference of squares, enhances your ability to factor and solve complex problems efficiently.