Problem 49 - Elementary Algebra

Question

For the following problems, factor the trinomials if possible. $$ 12 a^{2}+54 a-90 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial is \(6(2a - 3)(a + 5)\).

Step 1: Find the greatest common divisor (GCD) of the trinomial coefficients

Calculate the GCD of the coefficients 12, 54, and -90. The GCD in this case is 6.

Step 2: Simplify the trinomial

Divide each term of the trinomial by the GCD: $$ \frac{12a^2}{6} + \frac{54a}{6} - \frac{90}{6} = 2a^2 + 9a - 15 $$

Step 3: Factor the simplified trinomial

Use the product-sum method to find the factors of 2 and -15, such that their product is equal to the leading coefficient multiplied by the constant term, and their sum is equal to the middle coefficient: $$(2a)(a) = 2a^2$$ $$(-3)(5) = -15$$ $$(2a - 3)(a + 5) = 2a^2 + 9a - 15$$

Step 4: Write the final factored form

The factored trinomial is: $$ 12a^2 + 54a - 90 = 6(2a^2 + 9a - 15) = 6(2a - 3)(a + 5) $$

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is crucial when factoring trinomials. It simplifies expressions by finding the highest number that can divide all coefficients without leaving a remainder. This helps reduce complicated polynomials into simpler forms. Understanding the GCD is a stepping stone in polynomial simplification.
For example, consider the expression \(12a^2 + 54a - 90\). The coefficients are \(12\), \(54\), and \(-90\). To find the GCD:

• List the factors of each number:
• 12: 1, 2, 3, 4, 6, 12
• 54: 1, 2, 3, 6, 9, 18, 27, 54
• -90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

The largest common factor among these is \(6\). By dividing each term of the trinomial by \(6\), we simplify it, paving the way for further factoring.

Product-Sum Method

The product-sum method is a strategic approach to factor trinomials. It involves expressing the middle term as a combination of two numbers whose product is the product of the leading coefficient and the constant term.For the trinomial \(2a^2 + 9a - 15\), you start by multiplying the first and last coefficients (2 and -15):

• Product: \(2 \times -15 = -30\).

Now, find two numbers that multiply to \(-30\) and add to \(9\) (the middle coefficient). These numbers are \(-3\) and \(10\):

• \(-3 \times 10 = -30\)
• \(-3 + 10 = 9\)

The expression can be rewritten using these figures as \((2a - 3)(a + 5)\). This method is highly effective for quickly breaking down complex polynomials for further simplification.

Polynomial Simplification

Polynomial simplification involves reducing expressions to their simplest form by factoring. This process typically begins with identifying the GCD and using the product-sum method to break down trinomials. Let's illustrate this using \(12a^2 + 54a - 90\):

• Step 1: Identify the GCD, which is \(6\), and simplify to \(2a^2 + 9a - 15\).


• Step 2: Utilize the product-sum method to find factors, rewriting it as \((2a - 3)(a + 5)\).

By assembling these steps, we reach the simplest factored form: \(6(2a - 3)(a + 5)\). Simplification not only makes the expression more manageable but also enhances understanding of its structure and potential applications. Each phase of simplification sharpens one's algebraic skills and conveys deeper insights into polynomial behavior.