Problem 89 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ 4 a^{2}+a+9 $$

Step-by-Step Solution

Answer

If yes, provide the factored form. Answer: The trinomial $$4a^2 + a + 9$$ cannot be factored.

Step 1: Check for common factors

First, check if there are any common factors that can be factored out from the given trinomial. In this case, there are no common factors.

Step 2: Attempt to factor the trinomial using a product-sum method

Using the product-sum method, we try to find two numbers whose product is equal to the product of the leading coefficient and the constant term and whose sum is equal to the middle term coefficient (linear term coefficient). In this case, the product is: $$4 * 9 = 36$$ and the sum is: $$1$$ However, there are no two numbers that multiply to $$36$$ and add to $$1$$. Therefore, the trinomial cannot be factored using the product-sum method.

Step 3: Check for other factoring methods

Since factoring using the product-sum method is not possible, we check for other factoring methods such as difference of squares, perfect square trinomials or special trinomial patterns. However, the given trinomial does not fit any of these patterns.

Step 4: Conclusion

The given trinomial $$4a^2 + a + 9$$ cannot be factored.

Key Concepts

Product-Sum Method

The product-sum method is an approach used to factor trinomials. This method revolves around finding two numbers in a trinomial where their product equals the multiplication of the leading coefficient and the constant term. Additionally, their sum should equal the middle term's coefficient, which is responsible for the linear part of the trinomial.

To apply this method effectively, follow these steps:

• Multiply the leading coefficient by the constant term to get the product.
• Find two numbers whose product equals this calculated product and whose sum equals the middle term's coefficient.
• If you find such numbers, rewrite the middle term utilizing these two numbers, then factor by grouping.

In our exercise, the product generated was \(4 \times 9 = 36\) and the required sum is 1; however, no numbers satisfy both conditions simultaneously, making it impossible to factor using this method.

Leading Coefficient

When dealing with trinomials, the leading coefficient is the coefficient of the term with the highest degree—the quadratic term, typically. It's crucial in the product-sum method because it participates in the multiplication that leads to finding the necessary product.

In our example, the leading coefficient is 4, considering the expression \(4a^2 + a + 9\). Understanding the importance of the leading coefficient can also guide you in determining the complexity of factoring the trinomial.

If the leading coefficient is 1, such expressions are typically easier to factor using basic methods. However, a larger coefficient, like 4, can increase difficulty, requiring meticulous searching for the correct factor pairs.

Common Factors

Before diving into advanced factoring techniques, it's wise to check if a trinomial has common factors—a strategy that can simplify the expression conveniently. A common factor is a number or expression that divides each term of the trinomial evenly.

Checking for common factors comprises:

• Identifying any numerical or variable components shared among the trinomial's terms.
• Factoring out these components to simplify the expression, potentially leading to easier complete factoring.

For the given trinomial, \(4a^2 + a + 9\), there were no numerical or variable factors common to every term, meaning this step couldn't simplify our initial expression.

Trinomials

Trinomials are algebraic expressions composed of three terms, often appearing in the form of a quadratic expression, such as \(ax^2 + bx + c\). They can symbolize complex relationships and frequently appear in mathematical applications such as physics and economics.

In the context of our expression \(4a^2 + a + 9\), we see its structure highlighting:

• A quadratic term: \(4a^2\)
• A linear term: \(a\)
• A constant term: 9

Understanding these components is key to choosing the right factoring method and effectively simplifying or solving equations involving trinomials.

Factoring Methods

Factoring trinomials can involve various methods, each apt for different kinds of expressions and scenarios. While the product-sum method is a common choice, sometimes it's necessary to consider alternative approaches.

When a simple product-sum approach doesn't work, these are some strategies you can explore:

• Difference of Squares: Used when you have two perfect squares subtracted from one another.
• Perfect Square Trinomials: Applicable if the trinomial is the square of a binomial.
• Special Polynomial Patterns: Recognize patterns that flag specific forms, like cubes.

Unfortunately, in our trinomial \(4a^2 + a + 9\), none of these methods were applicable, underscoring that not all trinomials are factorable with standard approaches.