Hey guys! Let's dive into factoring this polynomial: p(x) = 3x³ - 20x² + 37x - 20. We already know that (x - 4) is one of its factors, which is a huge help. Our goal here is to rewrite p(x) as a product of linear factors, meaning we want to break it down into the form p(x) = (ax + b)(cx + d)(ex + f). It might seem intimidating at first, but don't worry, we'll break it down step by step, making it super easy to follow along. Factoring polynomials is a crucial skill in algebra and calculus, and mastering it will definitely boost your math game! We'll use polynomial division and some clever factoring techniques to get there. So, let's jump right in and get this polynomial factored!
Polynomial Division
Since we know (x - 4) is a factor, we can use polynomial division to find the quadratic factor. This method helps us divide the original polynomial by the known factor to find the remaining polynomial. Think of it like long division but with polynomials – it's a systematic way to break down the problem. By performing polynomial division, we're essentially asking, "How many times does (x - 4) fit into 3x³ - 20x² + 37x - 20?" The result will give us another polynomial that, when multiplied by (x - 4), gives us the original polynomial. This is a key step because it simplifies the problem from factoring a cubic polynomial to factoring a quadratic polynomial, which is generally easier to handle. Polynomial division is not just a mechanical process; it's a powerful tool for understanding the structure of polynomials and their factors. It's also essential for solving polynomial equations and understanding their roots. Once we've completed the division, we'll have a quadratic expression that we can factor further using techniques you might already be familiar with, such as factoring by grouping or using the quadratic formula. This step is crucial in our journey to expressing p(x) as a product of linear factors, so let's get to it!
Let's perform the polynomial division:
3x² - 8x + 5
x - 4 | 3x³ - 20x² + 37x - 20
- (3x³ - 12x²)
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-8x² + 37x
- (-8x² + 32x)
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5x - 20
- (5x - 20)
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0
This gives us a quotient of 3x² - 8x + 5.
Factoring the Quadratic
Now we need to factor the quadratic 3x² - 8x + 5. This step is all about breaking down the quadratic expression into two linear factors. There are several methods we can use, such as factoring by grouping, using the quadratic formula, or simply trial and error. Factoring by grouping involves finding two numbers that multiply to the product of the leading coefficient and the constant term (in this case, 3 * 5 = 15) and add up to the middle coefficient (-8). Once we find these numbers, we can rewrite the middle term and factor by grouping. Alternatively, the quadratic formula can be used to find the roots of the quadratic equation, which then allows us to write the quadratic in factored form. Trial and error involves intelligently guessing factors that might work and then checking if they multiply back to the original quadratic. Each method has its advantages, and the best one to use often depends on the specific quadratic we're dealing with. In this case, we'll use a method that feels most intuitive, but it's always good to be familiar with all the techniques. Factoring quadratics is a fundamental skill in algebra, and it's used extensively in many areas of mathematics. The ability to quickly and accurately factor a quadratic can save you a lot of time and effort when solving more complex problems. So, let's roll up our sleeves and get this quadratic factored!
We're looking for two numbers that multiply to 3 * 5 = 15 and add up to -8. Those numbers are -3 and -5. So, we can rewrite the quadratic as:
3x² - 3x - 5x + 5
Now, let's factor by grouping:
3x(x - 1) - 5(x - 1)
(3x - 5)(x - 1)
The Complete Factorization
Okay, we've done the heavy lifting! We started with a cubic polynomial and, through polynomial division and factoring a quadratic, we've broken it down into its linear factors. This is where everything comes together, and we express the original polynomial as a product of these factors. It’s like putting the pieces of a puzzle together to see the whole picture. This step is crucial because it gives us a complete understanding of the polynomial's structure. We can now easily identify the roots of the polynomial, which are the values of x that make the polynomial equal to zero. These roots are simply the values that make each of the linear factors equal to zero. The factored form also allows us to analyze the behavior of the polynomial function, such as its intercepts, turning points, and end behavior. Moreover, expressing a polynomial in factored form is essential for many applications in calculus, such as finding limits, derivatives, and integrals. So, take a moment to appreciate how we've transformed the original polynomial into a much more manageable form. Now, let's write out the complete factorization!
We know that p(x) = (x - 4)(3x² - 8x + 5), and we've factored the quadratic as (3x - 5)(x - 1). Therefore, the complete factorization of p(x) is:
p(x) = (x - 4)(3x - 5)(x - 1)
So, the final answer is:
CONTENT:
p(x) = (x - 4)(3x - 5)(x - 1)
REPAIR-INPUT-KEYWORD:
Rewrite p(x) = 3x³ - 20x² + 37x - 20 as a product of linear factors, given that (x-4) is a factor.
TITLE:
Factoring Polynomials How to Rewrite p(x) as Linear Factors
