Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(361 d^{2}-81\)

Step-by-Step Solution

Answer

The polynomial factors into \((19d - 9)(19d + 9)\).

Step 1: Identify Special Forms

The given polynomial is \(361d^2 - 81\). Recognize that this is a difference of squares problem, which takes the form \(a^2 - b^2\), where \(a = 19d\) and \(b = 9\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Substitute \(a = 19d\) and \(b = 9\) into this formula: \((19d)^2 - 9^2 = (19d - 9)(19d + 9)\).

Step 3: Write the Final Factored Form

The expression \(361d^2 - 81\) factors into \((19d - 9)(19d + 9)\). This is the final factored form of the polynomial.

Key Concepts

Difference of Squares

The "difference of squares" is a special factoring technique used in algebra to simplify certain types of polynomial expressions. This technique applies to expressions of the form \(a^2 - b^2\), where both \(a\) and \(b\) are squares. The reason it is called "difference of squares" is that it involves subtraction (difference) between two squared terms.

One of the key properties of these expressions is that they can be factored into two binomials: \((a - b)(a + b)\). This formula can be understood by expanding the factored form:

• The first terms \((a)(a)\) give \(a^2\).
• The outer \((-b)(a)\) and inner \((a)(b)\) terms cancel each other out since they sum up to zero.
• The last terms \((-b)(b)\) give \(-b^2\).

This result aligns with the initial expression \(a^2 - b^2\). Understanding this pattern is extremely useful, as it allows you to quickly identify and factor these kinds of expressions.

Polynomial Expressions

Polynomial expressions are sums of terms consisting of variables raised to whole number powers and possibly multiplied by coefficients. For instance, the expression \(361d^2 - 81\) is a polynomial with two terms. Each term is a product of a constant (coefficient) and a variable raised to a power.

Important key characteristics of polynomials include:

• The degree, which is the highest power of the variable (2 in \(361d^2\)).
• The leading coefficient, which is the coefficient of the term with the highest power (361 in this case).
• The constant term, which is any term without a variable (though it's absent in this polynomial).

Polynomials form the foundation for much of algebra and calculus and are important for expressing various mathematical operations. Factoring, in particular, aids in simplifying these expressions to solve equations.

Factoring Techniques

Factoring is a crucial algebraic process used to break down complex expressions into simpler, multipliable components. It is a foundational skill for solving equations and simplifying algebraic expressions.

Key factoring techniques include:

• Finding the Greatest Common Factor (GCF) to simplify terms.
• Recognizing Special Products like difference of squares or perfect square trinomials.
• Using systematic methods such as grouping or trial and error when applicable.

In the case of the expression \(361d^2 - 81\), recognizing it as a difference of squares allows for efficient factoring right away. Here, the application of the difference of squares formula results in the factored form \((19d - 9)(19d + 9)\). Mastery of these techniques enhances your ability to engage with more complex algebraic concepts.