Introduction
When we talk about factorization, we are essentially looking to break down a mathematical expression into its simplest possible form. In this case, we will delve into the factorization of the trinomial expression 3x^2 – 8x + 5. We will explore different methods and techniques used to factorize this expression and understand the underlying concepts involved.
The Trinomial Expression
Before we begin the factorization process, let’s first understand what the trinomial expression 3x^2 – 8x + 5 represents:
- 3x^2: This term signifies the coefficient 3 multiplied by x squared.
- -8x: This term represents the coefficient -8 multiplied by x (linear term).
- 5: This term is a constant term without any variables.
Factorization Methods
Factorization can be approached using various methods such as grouping, trial and error, completing the square, or even the quadratic formula. In the case of factoring trinomials, we commonly use methods like factoring by grouping or the AC method. Let’s explore how we can factorize the expression 3x^2 – 8x + 5 using these techniques:
Factoring by Grouping
In the factoring by grouping method, we group the terms in pairs and look for common factors that can be factored out. Here’s how we can apply this method to factorize 3x^2 – 8x + 5:
- Step 1: Multiply the coefficient of the x^2 term (3) by the constant term (5) to get 15.
- Step 2: Find two numbers that multiply to 15 and add up to the coefficient of the x term (-8). These numbers are -3 and -5.
- Step 3: Rewrite the middle term -8x as -3x – 5x.
- Step 4: Factor by grouping:
- 3x^2 – 3x – 5x + 5
- 3x(x – 1) – 5(x – 1)
- (3x – 5)(x – 1)
The AC Method
The AC method involves breaking down the trinomial into two binomials by considering the product of the leading coefficient (a) and the constant term (c). Let’s see how we can use the AC method to factorize 3x^2 – 8x + 5:
- Step 1: Multiply the leading coefficient (3) by the constant term (5) to get 15.
- Step 2: Find two numbers that multiply to 15 and add up to the middle coefficient (-8). These numbers are -3 and -5.
- Step 3: Rewrite the x term -8x as -3x – 5x.
- Step 4: Factor using the AC method:
- 3x^2 – 3x – 5x + 5
- 3x(x – 1) – 5(x – 1)
- (3x – 5)(x – 1)
Conclusion
Factorizing trinomial expressions like 3x^2 – 8x + 5 can be achieved through different methods such as factoring by grouping or using the AC method. By carefully examining the coefficients and constants, we can break down the expression into its simplest form of two binomials. Practice and familiarity with these methods are key to mastering factorization techniques in algebra.
