Problem 27 - College Algebra

Question

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

Step-by-Step Solution

Answer

The polynomial is factored as \((19d - 9)(19d + 9)\).

Step 1: Identify the Form

First, identify the structure of the given polynomial \(361 d^{2} - 81\). Notice that it resembles the form \(a^2 - b^2\), which is called a difference of squares.

Step 2: Express Each Term as a Square

Write each term in the polynomial as a square of another number: \(361 d^2 = (19d)^2\) and \(81 = 9^2\).

Step 3: Apply the Difference of Squares Formula

Recall the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Apply this formula by identifying \(a = 19d\) and \(b = 9\).

Step 4: Factor the Polynomial

Substitute \(a = 19d\) and \(b = 9\) into the difference of squares formula: \((19d - 9)(19d + 9)\). This is the factored form of the polynomial.

Key Concepts

Difference of Squares

The difference of squares is a specific type of polynomial expression. It's characterized by having only two terms, both perfect squares, separated by a subtraction sign. It takes the form \(a^2 - b^2\). Recognizing this pattern allows us to factor it efficiently.

• Both terms need to be perfect squares. For instance, in the expression \(361d^2 - 81\), \(361d^2\) is the square of \(19d\), and \(81\) is the square of \(9\).
• The key idea is to express these terms as squares: \((19d)^2 - (9)^2\).
• This expression fits the formula for a difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).

With these steps, you can break down complex-looking polynomials into simpler, factored forms.

Polynomial Expressions

Polynomial expressions involve variables raised to positive integer exponents. They are composed of several terms, which can include constants and variables. Understanding the structure of these expressions is crucial for operations like factoring.

• In a polynomial like \(361d^2 - 81\), terms are separated by addition or subtraction signs.
• The degree of the polynomial is the highest power of the variable present, in this case, 2, because of \(d^2\).
• A polynomial can often be rewritten in different forms to reveal useful properties, like factorizability.

Recognizing polynomials as sums of their individual terms allows for techniques like factoring to be applied.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They provide standard forms which simplify the manipulation and simplification of expressions.

• One of the most useful identities in algebra is the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).
• This identity is particularly helpful for factoring expressions quickly and accurately, if they match the structure.
• Applying these identities correctly transforms a seemingly complex polynomial into simpler products of binomials.

Knowing and applying these identities facilitates solving polynomial equations as well as simplifying expressions.