Problem 38 - Elementary and Intermediate Algebra

Question

Factor. If a polynomial can't be factored, write "prime." $$ x^{2}-9 $$

Step-by-Step Solution

Answer

The polynomial factors are \((x - 3)(x + 3)\).

Step 1: Recognize the type of polynomial

The given polynomial \(x^2 - 9\) is a difference of squares because it can be expressed as \(x^2 - 3^2\). Recall that any expression of the form \(a^2 - b^2\) can be factored as \((a - b)(a + b)\).

Step 2: Apply the difference of squares formula

Now apply the difference of squares formula to \(x^2 - 9\). Here, \(a = x\) and \(b = 3\), so \(x^2 - 9 = (x - 3)(x + 3)\).

Step 3: Verify the factorization

Multiply the factors \((x - 3)(x + 3)\) to ensure they yield the original polynomial. \((x - 3)(x + 3) = x^2 + 3x - 3x - 9 = x^2 - 9\), which shows our factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is an important pattern in algebra. It refers to expressions that take the form \(a^2 - b^2\). This pattern is special because it can always be factored into \((a - b)(a + b)\). Understanding this pattern is crucial when dealing with polynomial expressions that involve quadratic terms.

• "Difference" indicates subtraction: we are subtracting one square from another.
• "Squares" means we are working with square numbers or square expressions, like \(x^2\) or \(4^2\).

Recognizing a difference of squares allows you to factor polynomials efficiently. For example, if you have \(x^2 - 9\), you notice that it is a difference of squares because \(9\) is \(3^2\). This allows us to break down the expression into \((x - 3)(x + 3)\). The beauty of this method is in its simplicity and efficiency. By remembering this pattern, you can save time and avoid more complex factoring methods in many algebra problems.

Polynomial Expressions

Polynomial expressions consist of variables and constants combined using addition, subtraction, and multiplication. A polynomial is often written in terms of powers of a variable, like \(x\). Each individual term can be made up of a coefficient and a variable raised to a non-negative integer exponent. For example, the expression \(x^2 - 9\) is a polynomial.- **Terms of a Polynomial**: These are parts of a polynomial expression, such as \(x^2\) and \(-9\) in \(x^2 - 9\).- **Degree of a Polynomial**: This is determined by the highest power of the variable in the polynomial. In \(x^2 - 9\), the degree is 2 because of the \(x^2\) term.Polynomials are foundational in algebra because they model many natural processes and phenomena. Dealing with polynomial expressions requires recognizing patterns, like the difference of squares, and applying appropriate methods to simplify or solve the expressions.

Algebraic Factoring Methods

Algebraic factoring methods are strategies used to simplify polynomial expressions by breaking them into simpler parts, or factors. This process is important because it makes solving polynomial equations much easier. Factoring reveals the roots of the equation, helping to find solutions.There are several factoring methods, each useful based on the problem's structure:

• Greatest Common Factor (GCF): Factoring out the largest factor common to all terms.
• Difference of Squares: Used to factor expressions of the form \(a^2 - b^2\).
• Trinomials: Factoring expressions like \(ax^2 + bx + c\).

In our original exercise, identifying the expression \(x^2 - 9\) as a difference of squares was key to factoring it quickly into \((x - 3)(x + 3)\). These methods are not just techniques; they are tools that provide deeper insights into the structure of algebraic expressions. Mastering them opens the door to solving complex algebraic equations with ease.