Factoring Quadratics: Unveiling The Other Factor

Hey math enthusiasts! Today, we're diving into the world of factoring quadratics, specifically tackling a neat problem: If we know that ${3x^2 - 10x + 8}$ has a factor of ${3x - 4}$, what's the other factor? Don't worry, it's not as scary as it sounds. Factoring is like detective work for math problems. We're given a quadratic equation, which is just a fancy name for an equation with an ${x^2}$ term, and we're told one of its secret codes (a factor). Our mission? To crack the code and find the other one! This is a fundamental concept in algebra, and it's super useful for solving equations, understanding graphs, and lots of other mathy things. So, let's roll up our sleeves and get started. We'll explore the problem step-by-step, making sure everyone understands how to find the other factor. We'll use a combination of simple math and explanation to guide us through. Ready to become factoring masters? Let's go!

Understanding the Basics of Factoring

Alright, before we jump into the problem, let's get our bearings. Factoring is all about breaking down a mathematical expression into its building blocks, or factors. Think of it like taking a number, say 12, and figuring out what numbers you can multiply together to get 12. In this case, those numbers are the factors. For example, ${12 = 3 \cdot 4}$, so 3 and 4 are factors of 12. Similarly, in algebra, we factor expressions. When we factor a quadratic equation, like ${3x^2 - 10x + 8}$, we're trying to find two expressions that, when multiplied together, give us the original equation. These expressions are the factors. The reason this is useful is because if we can break down a complex expression into simpler factors, it often makes it easier to solve equations and understand the relationships between different parts of the expression. So, when the problem tells us ${3x - 4}$ is a factor, that's like giving us a head start in this factoring detective game. We already know one piece of the puzzle. Now, we just need to find the other.

The Relationship Between Factors and Quadratics

Understanding the relationship between factors and quadratics is like having a secret weapon. A quadratic equation, in its most general form, looks like ${ax^2 + bx + c}$, where a, b, and c are numbers. When we factor a quadratic, we're essentially writing it as a product of two binomials (expressions with two terms). For example, if we can factor ${3x^2 - 10x + 8}$, it would look like ${(px + q)(rx + s)}$, where p, q, r, and s are constants. When you multiply these two binomials, you'll get back to the original quadratic. This relationship is crucial because it allows us to break down a complex equation into smaller, more manageable parts. When the problem states that ${3x - 4}$ is a factor, it means that one of our binomials is ${3x - 4}$. So, we already have half of the solution. Our goal is to find the other binomial that, when multiplied by ${3x - 4}$, gives us the original quadratic. This whole process is often used for solving quadratic equations. By knowing the factors, we can easily find the roots or solutions of the equation, which are the values of x that make the equation equal to zero. This makes factoring a fundamental tool in algebra, helping us unlock the secrets of quadratic expressions and equations.

Solving the Problem: Finding the Other Factor

Now, let's put our knowledge to the test and actually solve the problem. We know that ${3x^2 - 10x + 8}$ has a factor of ${3x - 4}$. We can think of this problem in two main ways: using division or by using the educated guess. Since we are given one factor, the easiest approach is to divide the original quadratic by the given factor. This is because if ${3x - 4}$ is a factor of ${3x^2 - 10x + 8}$, then dividing ${3x^2 - 10x + 8}$ by ${3x - 4}$ should result in a clean division (no remainder), and the result of that division will be the other factor. Let's do the division: We set up the long division problem as follows: divide ${3x^2 - 10x + 8}$ by ${3x - 4}$. First, divide the leading term of the dividend (${3x^2}$) by the leading term of the divisor (${3x}$). This gives us ${x}$. We write ${x}$ as the first term of our quotient. Next, multiply ${x}$ by the entire divisor (${3x - 4}$). This gives us ${3x^2 - 4x}$. We subtract this from the dividend: ${(3x^2 - 10x + 8) - (3x^2 - 4x)}$, which simplifies to ${-6x + 8}$. Now, we bring down the next term, which is 8. We then divide the leading term of the new expression (${-6x}$) by the leading term of the divisor (${3x}$). This gives us ${-2}$. We write ${-2}$ as the next term in our quotient. Finally, we multiply ${-2}$ by the entire divisor (${3x - 4}$). This gives us ${-6x + 8}$. When we subtract this from ${-6x + 8}$, we get 0, indicating that the division is exact. So, the other factor is ${x - 2}$.

Alternative Method: Using the Educated Guess Approach

There's another cool way to solve this problem, which is sometimes quicker, especially if the coefficients are simple. We can use what's called the