Problem 70 - Elementary Algebra

Question

Factor. Either factor out the greatest common factor, factor by grouping, use the guess and check method, or use the \(a c\) method. $$ 3 x^{2}+9 x+3 $$

Step-by-Step Solution

Answer

3(x^2 + 3x + 1)

Step 1: Identify the greatest common factor (GCF)

Look at the coefficients of the terms in the polynomial. The coefficients are 3, 9, and 3. The greatest common factor of these coefficients is 3.

Step 2: Factor out the GCF

Factor out the greatest common factor, 3, from each term in the polynomial. This gives us: 3(x^2 + 3x + 1)

Step 3: Examine the remaining trinomial

The remaining trinomial inside the parentheses is x^2 + 3x + 1. There are no integers that multiply to 1 and add to 3, so the trinomial can't be factored further using integers.

Step 4: Write the final factored form

Since the trinomial x^2 + 3x + 1 cannot be factored further, the final factored form of the original polynomial is: 3(x^2 + 3x + 1)

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the highest number that divides exactly into two or more numbers. Finding the GCF is a crucial step in simplifying polynomials.
When you have a polynomial, you should first look at the coefficients of the terms. Coefficients are the numbers in front of the variables.
To find the GCF of 3, 9, and 3, identify the largest number that evenly divides all three. Here, it's 3.
You can factor out the GCF by dividing each term by this number, simplifying the expression.

Trinomial

A trinomial is a polynomial with three terms. It typically looks like this: \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients.
For example, in \(3x^2 + 9x + 3\), it's a trinomial because it has three terms: \(3x^2\), \(9x\), and \(3\).
Factoring trinomials often involves finding two numbers that multiply to give \(ac\) (the product of the coefficient of \(x^2\) and the constant term), and add up to the middle coefficient \(b\).
If such numbers can't be found, the trinomial may already be in its simplest form.

Factoring Methods

There are several common methods for factoring polynomials:

• Greatest Common Factor (GCF): As shown earlier, this involves factoring out the highest number common to all terms.
• Factoring by Grouping: This method is useful when a polynomial has four or more terms. Combine terms in pairs that can be factored further, then factor out the common binomial.
• Guess and Check Method: As the name suggests, this involves guessing pairs of factors until the correct pair is found. It's more trial and error but can be practical for simple trinomials.
• AC Method: Used for trinomials, especially when \(a\) is not equal to 1. Multiply \(a\) and \(c\), find two numbers that multiply to \(ac\) and add up to \(b\), then split the middle term and factor by grouping.

For the example \(3x^2 + 9x + 3\), the GCF method gives us \(3(x^2 + 3x + 1)\). The resulting trinomial \(x^2 + 3x + 1\) cannot be factored further using integers.
Thus, the final factored form remains as \(3(x^2 + 3x + 1)\).