Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{2}-9 $$

Short answer

(x - 3)(x + 3)

Step-by-step solution

Identify the type of polynomial

Recognize that the polynomial is in the form of a difference of squares. For a polynomial of the form \( a^2 - b^2 \), it can be factored using the rule \( a^2 - b^2 = (a - b)(a + b) \).

Express the polynomial as a difference of squares

In the given polynomial \( x^2 - 9 \), write 9 as a square: \( 9 = 3^2 \). Thus, the polynomial becomes \( x^2 - 3^2 \).

Apply the difference of squares formula

Using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \), substitute \( a = x \) and \( b = 3 \). Therefore, \( x^2 - 3^2 = (x - 3)(x + 3) \).

Key concepts

difference of squares

The difference of squares is a specific type of binomial that can be factored using a special formula. When you see an expression in the form of \(a^2 - b^2\), this is a difference of squares.
This concept is very useful as it lets us factor the polynomial into simpler binomials. The formula for factoring a difference of squares is: \[a^2 - b^2 = (a - b)(a + b) \]
In our exercise, we had \(x^2 - 9\). To recognize it as a difference of squares:

• Identify \(a^2 = x^2\), meaning \(a = x\)
• Identify \(b^2 = 9\), meaning \(b = 3\)

So, we see our polynomial is of the form \(a^2 - b^2\). Now, you can factor it by substituting \(a\) and \(b\):
\[ x^2 - 3^2 = (x - 3)(x + 3) \] Before moving forward with polynomial factorization, it's important to ensure we've correctly identified the form and applied the formula correctly.

algebraic expressions

An algebraic expression is a mathematical phrase that contains variables, numbers, and operators (like +, -, *, /). In our case, the given problem \(x^2 - 9\) is an algebraic expression containing a variable \(x\) and constants (numbers).
Polynomials are special kinds of algebraic expressions where only non-negative integer exponents of variables are involved.
The term 'algebraic expression' is useful to understand because:

• You can identify parts of the expression (terms, coefficients, variables).
• Helps to form strategies to manipulate the expression (like factoring).

For example, in \(x^2 - 9\):

• \(|| x^2 ||\) is the first term with coefficient 1 and exponent 2.
• \(|| -9 ||\) is a constant term, which can be seen as having an exponent of 0 for any variable.

Being familiar with algebraic expressions helps in polynomial factorization, as it provides a structured way to approach breaking down a polynomial into simpler parts.

factoring polynomials

Factoring polynomials is a process of expressing the polynomial as a product of its simpler factors. This is especially useful for simplifying equations, solving polynomial equations, and graphing polynomial functions.
There are several methods for factoring polynomials, including:

• Factoring out the greatest common factor (GCF).
• Using special formulas, like the difference of squares.
• Using the quadratic formula for second-degree polynomials.

In the exercise, we focused on the difference of squares, which is one of the most straightforward methods when applicable. Here’s a recap:

• We started with \(x^2 - 9\).
• Identified it as a difference of squares (\(x^2 - 3^2\)).
• Applied the difference of squares formula to factor it.

Therefore, \(x^2 - 9\) becomes \((x - 3)(x + 3)\). Understanding how to factor polynomials efficiently unlocks the ability to solve more complex algebraic equations and deepen your grasp of algebra.