Problem 34 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ 3 a^{2}+24 a+36 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial $$3a^2 + 24a + 36$$ is $$3(a + 2)(a + 6)$$.

Step 1: Look for common factors

First, check if all the terms have a common factor that can be factored out. In our trinomial, $$3a^2 + 24a + 36$$, the common factor is 3.

Step 2: Factor out the common factor

We will factor out the common factor (3) from the trinomial. So, we have: $$ 3(a^2 + 8a + 12) $$

Step 3: Factoring the quadratic expression

Now, we will factor the quadratic expression \((a^2 + 8a + 12)\) using the FOIL method. We need to find two binomials that when multiplied give us the quadratic expression \((a^2 + 8a + 12)\). To do this, we will find two numbers that multiply to give 12 (the constant term) and add up to 8 (the coefficient of the linear term). The numbers 2 and 6 fit this criteria.

Step 4: Write the factored form

As we have found the necessary factors, we can now write the factored form of the trinomial. We have: $$ 3(a^2 + 8a + 12) = 3(a + 2)(a + 6) $$ The trinomial $$3a^2 + 24a + 36$$ can be factored as $$3(a + 2)(a + 6)$$.

Key Concepts

Common Factors in Trinomials

When you're faced with factoring a trinomial, the first thing to check is whether the terms share a common factor. Finding common factors simplifies the expression, making the rest of the factoring process easier. In the trinomial given, \(3a^2 + 24a + 36\), each term can be divided by 3. Here’s the process:

• Look at each term: \(3a^2\), \(24a\), and \(36\).
• Ask what number multiplies into all these coefficients.
• Here, 3 is the greatest common factor (GCF).
• Factor 3 out of each term to simplify the trinomial to \(3(a^2 + 8a + 12)\).

By doing this, we've made it easier to handle the quadratic expression that remains.

Understanding Quadratic Expressions

A quadratic expression typically takes the form \(ax^2 + bx + c\), and its highest degree is the square of the variable. In our factored expression, \(a^2 + 8a + 12\), we have:

• \(a^2\) - The term with the highest power (the quadratic term).
• \(8a\) - The linear term.
• \(12\) - The constant term.

Understanding these parts helps us see how they contribute to the expressions' overall behavior. Quadratic equations are key in algebra because they describe a wide variety of phenomena and can often be solved by factoring.

Breaking Down Binomials

A binomial is an algebraic expression with two terms, like \((a + 2)\) or \((a + 6)\). In our problem, we needed to factor the quadratic expression into two binomials. This involves finding two numbers

• that multiply to the constant term (12)
• and also add up to the coefficient of the linear term (8)

These numbers are 2 and 6. Thus, the quadratic expression \(a^2 + 8a + 12\) can be factored into the binomials \((a + 2)(a + 6)\). Understanding how binomials function is crucial since they often form the building blocks of more complex algebraic expressions.

Using the FOIL Method for Verification

The FOIL method is a handy tool for multiplying two binomials. It stands for First, Outside, Inside, Last, signifying the order in which you multiply the terms:

• First: Multiply the first terms in each binomial. For \((a + 2)(a + 6)\), that's \(a \cdot a = a^2\).
• Outside: Multiply the outer terms: \(a \cdot 6 = 6a\).
• Inside: Multiply the inner terms: \(2 \cdot a = 2a\).
• Last: Multiply the last terms in each binomial: \(2 \cdot 6 = 12\).

When added together, these give you \(a^2 + 6a + 2a + 12 = a^2 + 8a + 12\). The FOIL method confirms that \((a + 2)(a + 6)\) indeed multiplies back to the original quadratic expression. It’s an excellent technique to ensure your factored expression is correct.