Question

Factor the difference of two squares. $$ 36 x^{2}-9 $$

Short answer

(6x-3)(6x +3)

Step-by-step solution

Identify the Difference of Squares

Recognize that the given expression is in the form of a difference of squares, which follows the pattern:

Write Each Term as a Square

Re-write each term as a square: For the term 36x^2, notice that 36x^2 = (6x)^2 5. For the term 9, notice that 9 = 3^2

Apply the Difference of Squares Formula

Use the formula a^2 - b^2 = (a - b)(a + b) . Here A = 6x and B = 3. Substitute these values into the formula: (6x)^2 -3^2= (6x- 3)(6x+3)

Key concepts

Understanding Factoring

Factoring is an essential skill in algebra. It involves breaking down an expression into simpler components that can be multiplied to produce the original expression. One common method of factoring is by recognizing patterns, such as the difference of squares. The difference of squares occurs when you have an expression like \(a^2 - b^2\).
This can be factored using the formula: \(a^2 - b^2 = (a - b)(a + b)\).
In our example, we start with \(36x^2 - 9\). By identifying 36x^2 and 9 as squares (since \(36x^2 = (6x)^2\) and \(9 = 3^2\)), we can apply the difference of squares formula to factor the expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They don't have equality signs, unlike equations. In the expression \(36x^2 - 9\), we see a combination of constants (numbers) and variables (letters representing numbers). Recognizing patterns in algebraic expressions is key to simplifying them.
When working with algebraic expressions:

• Look for common factors.
• Identify patterns like the difference of squares.
• Simplify the expression by factoring.
  Using these techniques helps in identifying how to break down and work with complex expressions easily.

In our exercise, we broke down each term \(36x^2\) and \(9\) into their square components to factor the entire expression.

Understanding Polynomial Identities

Polynomial identities are equations that are true for all values of the variable involved. They help us to transform and simplify polynomial expressions. A crucial polynomial identity in this context is the difference of squares.
For any real numbers \(a\) and \(b\), the formula is:

• \(a^2 - b^2 = (a - b)(a + b)\)

In our exercise, \(a = 6x\) and \(b = 3\). Substituting these values into the difference of squares identity, we get:

• \(36x^2 - 9 = (6x - 3)(6x + 3)\)

This identity helps in quickly and correctly factoring the expression without needing to perform long multiplications.
Understanding polynomial identities can significantly ease solving many algebraic expressions and equations.