Factoring The Trinomial A^2 + 10a + 16 Step-by-Step

Introduction

In the realm of algebra, factoring trinomials is a fundamental skill that unlocks the door to solving quadratic equations and simplifying expressions. This article delves into the process of factoring the trinomial a^2 + 10a + 16, providing a step-by-step guide and exploring the underlying concepts. We will not only demonstrate how to factor this specific trinomial but also equip you with the knowledge to tackle similar problems with confidence. Understanding factoring is crucial for various mathematical applications, from solving algebraic equations to simplifying complex expressions in calculus and beyond. Let's embark on this journey to master the art of factoring!

Understanding Trinomials and Factoring

Before we dive into the specifics of factoring a^2 + 10a + 16, it's essential to grasp the basics of trinomials and the concept of factoring itself. A trinomial is a polynomial expression consisting of three terms. In the general form of a quadratic trinomial, we often see expressions like ax^2 + bx + c, where a, b, and c are constants, and x is the variable. Our trinomial, a^2 + 10a + 16, fits this form, with a = 1, b = 10, and c = 16. Factoring, in its essence, is the reverse process of multiplication. It involves breaking down an expression into a product of its factors. Think of it as unwrapping a gift to reveal its components. For instance, the number 12 can be factored into 2 × 6 or 3 × 4. Similarly, we aim to express the trinomial as a product of two binomials (expressions with two terms). The goal of factoring a trinomial is to rewrite it as a product of two binomials. This process is crucial for simplifying expressions, solving quadratic equations, and understanding the behavior of polynomial functions. Mastering factoring techniques not only helps in solving mathematical problems but also enhances analytical thinking and problem-solving skills, which are valuable in various fields beyond mathematics.

The Factoring Process: A Step-by-Step Approach

Now, let's embark on the journey of factoring the trinomial a^2 + 10a + 16. We'll break down the process into manageable steps, making it easy to follow and understand. This structured approach will empower you to tackle similar factoring problems with confidence and clarity.

Step 1: Identify the Coefficients

The first step is to identify the coefficients in the trinomial. In a^2 + 10a + 16, the coefficient of the a^2 term is 1, the coefficient of the a term is 10, and the constant term is 16. These coefficients play a crucial role in determining the factors.

Step 2: Find Two Numbers

The core of factoring this type of trinomial lies in finding two numbers that satisfy two conditions: they must multiply to the constant term (16 in this case) and add up to the coefficient of the a term (10 in this case). This step involves some mental math and strategic thinking. We need to consider pairs of numbers that multiply to 16, such as 1 and 16, 2 and 8, and 4 and 4. Then, we check which of these pairs also add up to 10. In this instance, the pair 2 and 8 fits the bill perfectly, as 2 × 8 = 16 and 2 + 8 = 10.

Step 3: Construct the Binomial Factors

Once we've identified the numbers 2 and 8, we can construct the binomial factors. We create two binomials, each starting with a (since the trinomial's leading term is a^2). The two numbers we found, 2 and 8, will be the constant terms in these binomials. Thus, we form the factors (a + 2) and (a + 8). The signs within the binomials are determined by the signs in the original trinomial. Since both the a term and the constant term are positive, we use addition in both binomials.

Step 4: Verify the Solution

The final step is to verify that our factored form is correct. We can do this by multiplying the two binomials (a + 2) and (a + 8) using the FOIL method (First, Outer, Inner, Last) or the distributive property. This will expand the binomials and should result in the original trinomial if our factoring is correct.

Verifying the Solution: Expanding the Binomials

To ensure the accuracy of our factored form, we must verify that the product of the binomials (a + 2) and (a + 8) indeed equals the original trinomial, a^2 + 10a + 16. This verification process not only confirms our solution but also deepens our understanding of the factoring process itself.

Using the FOIL Method

The FOIL method is a mnemonic acronym that stands for First, Outer, Inner, Last, representing the order in which we multiply the terms of the two binomials:

• First: Multiply the first terms of each binomial: a × a = a^2
• Outer: Multiply the outer terms of the binomials: a × 8 = 8a
• Inner: Multiply the inner terms of the binomials: 2 × a = 2a
• Last: Multiply the last terms of each binomial: 2 × 8 = 16

Now, we add these products together: a^2 + 8a + 2a + 16. Combining the like terms (8a and 2a), we get a^2 + 10a + 16, which is precisely the original trinomial.

Using the Distributive Property

Alternatively, we can use the distributive property to expand the binomials. This involves multiplying each term in the first binomial by each term in the second binomial:

• a × (a + 8) = a^2 + 8a
• 2 × (a + 8) = 2a + 16

Adding these results together, we get (a^2 + 8a) + (2a + 16) = a^2 + 10a + 16. Again, this confirms that our factored form is correct.

The Solution

After successfully identifying the factors and verifying our solution, we can confidently state that the factored form of the trinomial a^2 + 10a + 16 is (a + 2)(a + 8). This solution represents the trinomial expressed as a product of two binomials, which is the essence of factoring.

Conclusion

Factoring the trinomial a^2 + 10a + 16 is a prime example of how algebraic expressions can be broken down into simpler components. By following a systematic approach, we identified the key numbers, constructed the binomial factors, and verified our solution. This process not only solves the specific problem at hand but also equips us with valuable skills for tackling a wide range of factoring challenges. Mastering factoring techniques is crucial for success in algebra and beyond, as it forms the foundation for solving equations, simplifying expressions, and understanding more advanced mathematical concepts. So, embrace the challenge, practice the steps, and unlock the power of factoring!

Keywords

Factoring trinomials, quadratic expressions, algebraic factorization, binomial factors, FOIL method, distributive property, solving equations.

Is the trinomial a^2+10a+16 not factorable?