Problem 66 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ c^{2}+6 c+9 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial $$c^2+6c+9$$ is $$(c+3)^2$$.

Step 1: Identify the trinomial

The given trinomial is a quadratic equation in the form of $$ax^2+bx+c$$, where a = 1, b = 6, and c = 9.

Step 2: Find two numbers that multiply to c (9) and add up to b (6)

We need to find two numbers that multiply to 9 and add up to 6. After trying different combinations, we find that the numbers 3 and 3 satisfy the conditions: $$(3)(3)=9$$ $$3+3=6$$

Step 3: Rewrite the trinomial in factored form

Now that we have the numbers, let's use them to write the trinomial in its factored form. The factored form of the trinomial will be $$(c+3)(c+3)$$

Step 4: Simplify the factored form

Since both factors are the same, we can write the simplified expression as $$(c+3)^2$$ Thus, the factored form of the given trinomial $$c^2+6c+9$$ is $$(c+3)^2$$.

Key Concepts

Trinomials

A trinomial is a type of polynomial consisting of three distinct terms. Typically, these expressions are represented in the standard quadratic form: \(ax^2 + bx + c\). Trinomials are quite common in algebraic applications and are known for their role in various mathematical equations and functions.
Trinomials arise in numerous mathematical problems and real-life scenarios. They are often used in physics, engineering, and economics to model complex relationships.
In this particular case, the example \(c^2 + 6c + 9\) clearly shows a trinomial where each part serves a specific purpose:

• The term \(c^2\) is the squared variable representing the quadratic aspect of the polynomial.
• The linear term, \(6c\), reflects the influence of the variable \(c\) on the expression.
• The constant, 9, adds a specific value to complete the expression.

Understanding the structure of trinomials is crucial for factoring them effectively.

Factoring Techniques

Factoring is a fundamental algebraic process used to simplify polynomial expressions like trinomials. When you factor an expression, you are essentially breaking it down into a product of its simpler components. This approach can make solving algebraic equations much easier.
There are several techniques to factor trinomials, but one of the most accessible is trial and error, especially when dealing with the standard form of \(ax^2 + bx + c\). The goal is to find two numbers that multiply to \(c\) (the constant) and add up to \(b\) (the coefficient of \(x\)).
In the example exercise, the trinomial \(c^2 + 6c + 9\) was factored by finding two numbers that multiply to 9 and add up to 6—these numbers were 3 and 3.

• First, consider pairs of factors of the constant term (9): (1, 9) and (3, 3).
• Next, check which pair sums up to the linear coefficient (6). Here, (3, 3) works.
• Finally, rewrite the trinomial as a product of binomials using these numbers: \((c+3)(c+3)\).

Simplifying gives the final factorization: \((c + 3)^2\). Using this strategy helps you solve complex problems by breaking them into easier, manageable parts.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are foundational in algebra and serve as building blocks for forming equations and functions.
Expressions like \(c^2 + 6c + 9\) involve variables (letters like \(c\)) to represent unknown or varying quantities. Through the process of simplification and manipulation, you often find relationships or solutions to problems these expressions pose.
There are different parts of an algebraic expression that play unique roles:

• Each term, such as \(c^2\), carries a variable raised to a power, called an exponent.
• Coefficients, like 6, are numbers multiplied by the variable (\(c\)).
• Constants, such as 9, are independent of any variables and add fixed values to the expression.

Understanding these components helps you interpret and manipulate expressions effectively. Once you grasp how to break down and factor algebraic expressions, like with the trinomial shown, you gain a powerful toolset for tackling a wide range of mathematical challenges.