Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of polynomials, specifically focusing on how to break down a cubic polynomial into its linear factors. We'll be tackling the polynomial p(x) = 5x³ - 44x² + 61x + 14. The cool thing is, we already know that (x - 7) is a factor. This is a huge head start, and we're going to use this information to unravel the complete factorization of p(x). So, buckle up and let's get started!
Utilizing the Known Factor (x - 7)
Our adventure begins with the crucial piece of information: (x - 7) is a factor of p(x). What does this actually mean? It means that if we divide p(x) by (x - 7), we'll get a clean division with no remainder. This is a fundamental concept in polynomial factorization, guys. The result of this division will be another polynomial, a quadratic in this case, which will hold the key to finding the remaining factors. We'll use polynomial long division (or synthetic division, if you're a fan) to perform this division. Polynomial long division might seem a bit intimidating at first, but it's a systematic process that breaks down the problem into manageable steps. Think of it like long division with numbers, but instead of digits, we're working with terms involving 'x'. The goal is to find a quotient polynomial and a remainder. Since we know (x - 7) is a factor, we expect the remainder to be zero, which will confirm our initial knowledge and give us the quotient polynomial that we can then further factorize.
When we perform the division, we're essentially asking: "How many times does (x - 7) fit into 5x³ - 44x² + 61x + 14?" We start by looking at the highest degree terms. How many times does x fit into 5x³? The answer is 5x². So, we write 5x² above the division bar and multiply it by (x - 7), which gives us 5x³ - 35x². We then subtract this from the original polynomial, and the process continues. This iterative process narrows down the polynomial until we obtain the quotient which will be the other factor of the cubic polynomial. After carefully executing the polynomial long division (or synthetic division, if that’s your preference), we arrive at a quotient of 5x² - 9x - 2. This is awesome! It means we can now rewrite p(x) as (x - 7)(5x² - 9x - 2). We've successfully factored out the linear factor (x - 7), and we're left with a quadratic expression. But our journey isn't over yet; we need to factor this quadratic as well.
Factoring the Quadratic 5x² - 9x - 2
Now we're faced with the quadratic expression 5x² - 9x - 2. Factoring quadratics can sometimes feel like solving a puzzle, but there are a few techniques we can use. One common method is to look for two numbers that multiply to give the product of the leading coefficient (5) and the constant term (-2), which is -10, and add up to the middle coefficient (-9). Let's think about the factors of -10. We have 1 and -10, -1 and 10, 2 and -5, and -2 and 5. Bingo! The pair 1 and -10 fits the bill: 1 * -10 = -10 and 1 + (-10) = -9. This technique allows us to break down the middle term and rewrite the quadratic in a way that we can factor by grouping. So, we rewrite 5x² - 9x - 2 as 5x² + x - 10x - 2. Now we can group the first two terms and the last two terms: (5x² + x) + (-10x - 2). From the first group, we can factor out an x, leaving us with x(5x + 1). From the second group, we can factor out a -2, leaving us with -2(5x + 1). Notice that both groups now have a common factor of (5x + 1). This is exactly what we wanted! We can factor out (5x + 1), which gives us (5x + 1)(x - 2). Awesome! We've successfully factored the quadratic 5x² - 9x - 2 into two linear factors.
The Complete Factorization
We've come a long way, guys! We started with a cubic polynomial and a known factor, and we've systematically broken it down into its linear constituents. We first used the known factor (x - 7) to divide the original polynomial, resulting in the quadratic 5x² - 9x - 2. Then, we skillfully factored this quadratic into (5x + 1)(x - 2). Now, we can put it all together! The complete factorization of p(x) = 5x³ - 44x² + 61x + 14 is therefore (x - 7)(5x + 1)(x - 2). This is the final answer! We have successfully expressed the cubic polynomial as a product of three linear factors. This means we've found the roots of the polynomial, which are the values of x that make p(x) equal to zero. These roots are 7, -1/5, and 2. Factoring polynomials into linear factors is a fundamental skill in algebra and calculus. It allows us to solve polynomial equations, find the roots of functions, and analyze the behavior of polynomial graphs. Mastering these techniques opens up a whole new world of mathematical possibilities, and this process can be applied to a wide range of polynomial problems. So keep practicing, keep exploring, and keep unlocking the secrets of the mathematical universe!
Expressing p(x) as a Product of Linear Factors
After our detailed journey, we've arrived at the final destination! We can now express p(x) as a product of linear factors. Based on our work above, the answer is:
p(x) = (x - 7)(5x + 1)(x - 2)
And there you have it! We've successfully factored the polynomial p(x) into its linear components. This process not only demonstrates the power of polynomial division and quadratic factorization but also highlights the interconnectedness of different algebraic concepts. Remember, practice makes perfect, so keep exploring these techniques and challenging yourself with new problems. The world of polynomials is vast and fascinating, and the more you delve into it, the more you'll discover!
