Problem 26 - Elementary Algebra

Question

Factor the following problems, if possible. $$ 9 a^{2}-7 a+2 $$

Step-by-Step Solution

Answer

Answer: The quadratic equation \(9a^2 - 7a + 2\) can be factorized as \((3a + 1)(3a + 2)\).

Step 1: Identify the terms of the quadratic equation

In the given quadratic equation, the three terms to consider are: - First term: \(9a^2\) - Second term: \(-7a\) - Third term: \(2\)

Step 2: Search for the greatest common divisor

Determine if there is a greatest common divisor among all the terms. If so, factor it out. In this case, there is no common divisor greater than 1 for all three terms, so no factoring can be done in this step.

Step 3: Apply the trial and error method

Since there is no common divisor among the terms, the trial and error method can be applied to factorize the quadratic equation. In this method, we try different combinations of factors of the first term and the last term to create two binomials that multiply to give the given quadratic equation. For the first term \(9a^2\), its factors are \((a)(9a)\) and \((3a)(3a)\). For the third term, \(2\), its factors are \((1)(2)\) and \((2)(1)\). Now, we need to create binomials by combining the factors of the first term and the third term and applying the FOIL (First, Outer, Inner, Last method). In order to form the original quadratic equation, the Inner plus Outer products must equal the middle term \(-7a\).

Step 4: Test different combinations of factors

Let's try different combinations to find the correct pair of binomials. 1. Trying the combination \((a + 1)(9a + 2)\). Upon applying the FOIL method, we have \(9a^2 + 2a + 9a + 2\), which is not equal to the original quadratic equation. 2. Trying the combination \((a + 2)(9a + 1)\). Upon applying the FOIL method, we have \(9a^2 + a + 18a + 2\), which is not equal to the original quadratic equation. 3. Trying the combination \((3a + 1)(3a + 2)\). Upon applying the FOIL method, we have \(9a^2 + 6a + 3a + 2\), which equals the original quadratic equation.

Step 5: Write the factored solution

The correct factored solution is \((3a + 1)(3a + 2)\). So, the quadratic equation \(9a^2 - 7a + 2\) can be factorized as \((3a + 1)(3a + 2)\).

Key Concepts

Trial and Error Method

When factoring quadratic equations, especially when there is no greatest common divisor (GCD) to simplify the equation, the trial and error method can be a reliable approach. This method involves selecting combinations of factors from the first and last terms of the quadratic, in order to construct two binomial expressions.

For example, in the quadratic equation \(9a^2 - 7a + 2\), the possible factor pairs for \(9a^2\) are \((a)(9a)\) and \((3a)(3a)\), while those for \(2\) are \((1)(2)\) and \((2)(1)\). We attempt different combinations, checking which pair of binomials satisfactorily expands to match the original quadratic equation.

A combination is a winner if the Inner and Outer products provide the linear term of the original quadratic equation. This may involve multiple attempts, hence the name "trial and error." It's crucial to work systematically and carefully check each pairing.

FOIL Method

The FOIL method is an essential tool in algebra for expanding the product of two binomials and is particularly helpful when using the trial and error method. FOIL stands for "First, Outer, Inner, Last," referring to the order in which you multiply the terms in each binomial.

Consider two binomials, such as \((3a + 1)\) and \((3a + 2)\). Applying FOIL, you multiply as follows:

• **First**: Multiply the first terms of each binomial: \(3a \cdot 3a = 9a^2\)
• **Outer**: Multiply the outer terms: \(3a \cdot 2 = 6a\)
• **Inner**: Multiply the inner terms: \(1 \cdot 3a = 3a\)
• **Last**: Multiply the last terms: \(1 \cdot 2 = 2\)

Adding these together gives: \(9a^2 + 6a + 3a + 2\), which simplifies to \(9a^2 + 9a + 2\).

Notice here that the combined middle terms \((6a + 3a)\) must match the middle term of the original quadratic. Verifying each attempt with FOIL ensures that your binomials are correctly paired.

Greatest Common Divisor

The greatest common divisor (GCD) is a valuable concept when simplifying and factoring polynomial expressions. It represents the largest number (or term) that divides all terms within a polynomial without leaving a remainder.

In the problem \(9a^2 - 7a + 2\), determining the GCD involves checking if there is a number or polynomial that commonly divides the coefficients and terms.

• \(9a^2\) has factors 1, 3, 9, \(a\)
• \(-7a\) has factors 1, 7, \(-7\), \(a\)
• \(2\) has factors 1, 2

Since the only common factor is 1, the GCD is 1, indicating that there is no greater factor shared among all terms. When there is no GCD greater than 1, factoring must proceed with methods like trial and error without initial simplification by common factors.