Problem 21 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$9 x^{2}-36$$

Step-by-Step Solution

Answer

The polynomial factors as \(9(x - 2)(x + 2)\).

Step 1: Finding the Greatest Common Factor (GCF)

First, we need to identify if there is a greatest common factor (GCF) in the polynomial \(9x^2 - 36\). Both terms, \(9x^2\) and \(36\), are divisible by 9. Thus, the GCF is 9.

Step 2: Factoring Out the GCF

Next, we factor the GCF out of the polynomial. This is done by dividing each term of the polynomial by the GCF. So, \(9x^2 - 36\) becomes \(9(x^2 - 4)\).

Step 3: Recognizing a Difference of Squares

Now, look at the trinomial inside the parentheses, \(x^2 - 4\). This is a difference of squares, as \(x^2\) is a perfect square and 4 is \(2^2\), another perfect square. The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\).

Step 4: Applying the Difference of Squares Formula

Using the formula \(a^2 - b^2 = (a - b)(a + b)\), where \(a = x\) and \(b = 2\), we factor \(x^2 - 4\) as \((x - 2)(x + 2)\).

Step 5: Writing the Complete Factored Form

Finally, bring the GCF back into the factored expression. The fully factored form of \(9x^2 - 36\) is therefore \(9(x - 2)(x + 2)\).

Key Concepts

greatest common factor

When you start factoring any polynomial, the first step is to look for the Greatest Common Factor (GCF). This just means finding the largest number or expression that divides each term of the polynomial. It's important because taking out the GCF can make the rest of the factoring process a lot simpler.
For example, in the polynomial \(9x^2 - 36\), both terms, \(9x^2\) and \(36\), share the number 9 as a factor. So, 9 is the GCF here.

• Step 1: Check each term of the polynomial.
• Step 2: Determine the highest number that divides all of them.
• Step 3: This number is your GCF.

Once you identify the GCF, divide each term of the polynomial by this number and factor it out. It simplifies everything and sets you up for the next steps in the factoring process.

difference of squares

The concept of a "Difference of Squares" is another key tool in factoring. This method applies when you have a polynomial of the form \(a^2 - b^2\).
The expression can be rewritten as \( (a - b)(a + b) \), which tells us that it's all about breaking down into two smaller terms. For our example, after factoring out the GCF from \(9x^2 - 36\), you're left with \(x^2 - 4\). Here, \(x^2\) is a perfect square, and 4 is \(2^2\), another perfect square. Recognizing both terms as squares lets us apply the difference of squares formula.
To do this:

• Identify \(a = x\) and \(b = 2\).
• Plug these into the formula \(a^2 - b^2 = (a - b)(a + b)\).

Now, you can express \((x^2 - 4)\) as \((x - 2)(x + 2)\).
This makes the polynomial much simpler to work with and solves it efficiently!

factoring polynomials

Factoring polynomials involves breaking a complicated expression into simpler components (or factors) that, when multiplied together, give the original polynomial. It helps in solving equations and simplifies expressions for further work.
The overall steps include identifying the GCF and looking for patterns like the difference of squares.
Let's summarize the process with the polynomial \(9x^2 - 36\):

• Identify the GCF: It's 9 in this case, so you start with \(9(x^2 - 4)\).
• Recognize any patterns: Here, \(x^2 - 4\) is a difference of squares.
• Apply formulas: Use the difference of squares pattern \((x^2 - 4) = (x - 2)(x + 2)\).

Combine everything, and you have the fully factored form \(9(x - 2)(x + 2)\). This process breaks down polynomials into their simplest factors, making them easier to work with and understand.