Factoring Polynomials Step-by-Step Solution For 2x³ - 3x² - 3x + 2
Polynomial factorization is a fundamental concept in algebra, and mastering it is crucial for solving various mathematical problems. In this comprehensive guide, we will delve into the process of factoring polynomials, with a specific focus on the polynomial 2x³ - 3x² - 3x + 2. We will explore different techniques and strategies to break down this polynomial into its constituent factors, providing you with a solid understanding of the underlying principles. Factoring polynomials involves expressing a polynomial as a product of simpler expressions. These simpler expressions are called factors. The process of factoring is essentially the reverse of the distributive property. Understanding how to factor is essential for simplifying algebraic expressions, solving equations, and analyzing graphs of polynomial functions. In particular, this discussion tackles the polynomial 2x³ - 3x² - 3x + 2, offering a step-by-step approach to finding its factors. Factoring can simplify complex expressions, making them easier to work with. For instance, factoring allows us to simplify rational expressions, solve polynomial equations, and understand the behavior of polynomial functions. Different methods exist for factoring, including factoring out the greatest common factor (GCF), factoring by grouping, using the difference of squares formula, and employing the sum or difference of cubes formulas. The choice of method depends on the structure of the polynomial. When factoring polynomials, it's essential to follow a systematic approach. Start by looking for a GCF, then consider other methods like grouping or special formulas. Practice is crucial to develop proficiency in factoring. By working through various examples and problems, you'll become more adept at recognizing patterns and applying the appropriate factoring techniques. Understanding polynomial factorization is not just an academic exercise; it has practical applications in various fields, including engineering, physics, and computer science. For example, factoring is used in cryptography, data compression, and image processing. This guide aims to equip you with the knowledge and skills necessary to confidently tackle polynomial factorization problems. By mastering these techniques, you'll enhance your problem-solving abilities and gain a deeper appreciation for the elegance and power of algebra.
Identifying a Factor: The Factor Theorem
In the given problem, we are told that (x - 2) is one factor of the polynomial 2x³ - 3x² - 3x + 2. This information is a crucial starting point for our factorization journey. This is where the Factor Theorem becomes invaluable. The Factor Theorem is a powerful tool that helps us identify factors of polynomials. It states that if a polynomial f(x) has a factor (x - a), then f(a) = 0. Conversely, if f(a) = 0, then (x - a) is a factor of f(x). In simpler terms, if we substitute x = a into the polynomial and the result is zero, then (x - a) is a factor. Let's apply this theorem to our polynomial, f(x) = 2x³ - 3x² - 3x + 2. Since we are given that (x - 2) is a factor, we can verify this by substituting x = 2 into the polynomial: f(2) = 2(2)³ - 3(2)² - 3(2) + 2 = 16 - 12 - 6 + 2 = 0. As expected, f(2) = 0, confirming that (x - 2) is indeed a factor. Now that we know (x - 2) is a factor, our goal is to find the other factor(s) that, when multiplied by (x - 2), will give us the original polynomial 2x³ - 3x² - 3x + 2. We have several methods at our disposal to achieve this, including polynomial long division and synthetic division. These methods allow us to divide the polynomial by the known factor (x - 2) and obtain the quotient, which will represent the remaining factor(s). Understanding and applying the Factor Theorem is essential for efficient polynomial factorization. It allows us to quickly identify potential factors and simplify the factorization process. By combining the Factor Theorem with other techniques like polynomial division, we can effectively break down complex polynomials into their constituent factors.
Polynomial Long Division: Unveiling the Remaining Factors
Now that we've confirmed (x - 2) is a factor, let's employ polynomial long division to find the other factor(s) of 2x³ - 3x² - 3x + 2. Polynomial long division is a systematic method for dividing polynomials, similar to the long division method used for dividing numbers. It allows us to divide a polynomial by another polynomial of equal or lower degree. To perform polynomial long division, we set up the division problem as follows:
x - 2 | 2x³ - 3x² - 3x + 2
We begin by dividing the leading term of the dividend (2x³) by the leading term of the divisor (x), which gives us 2x². This is the first term of our quotient. We then multiply the divisor (x - 2) by 2x² and subtract the result from the dividend:
2x²
x - 2 | 2x³ - 3x² - 3x + 2
-(2x³ - 4x²)
-----------------
x² - 3x
Next, we bring down the next term of the dividend (-3x) and repeat the process. We divide the leading term of the new dividend (x²) by the leading term of the divisor (x), which gives us x. This is the second term of our quotient. We multiply the divisor (x - 2) by x and subtract the result:
2x² + x
x - 2 | 2x³ - 3x² - 3x + 2
-(2x³ - 4x²)
-----------------
x² - 3x
-(x² - 2x)
-----------
-x + 2
We bring down the last term of the dividend (+2) and repeat the process one more time. We divide the leading term of the new dividend (-x) by the leading term of the divisor (x), which gives us -1. This is the third term of our quotient. We multiply the divisor (x - 2) by -1 and subtract the result:
2x² + x - 1
x - 2 | 2x³ - 3x² - 3x + 2
-(2x³ - 4x²)
-----------------
x² - 3x
-(x² - 2x)
-----------
-x + 2
-(-x + 2)
-----------
0
The remainder is 0, which confirms that (x - 2) is indeed a factor. The quotient we obtained, 2x² + x - 1, represents the other factor of the polynomial. Therefore, we have successfully factored the polynomial as (x - 2)(2x² + x - 1). Polynomial long division provides a structured approach to dividing polynomials, allowing us to identify the remaining factors after we know one factor. This method is particularly useful when dealing with higher-degree polynomials. The result of the long division gives us the other factor, which is 2x² + x - 1. This quadratic expression can be further factored, if possible, to obtain the complete factorization of the original polynomial.
Factoring the Quadratic Factor: Completing the Factorization
After performing polynomial long division, we've arrived at the quadratic factor 2x² + x - 1. To fully factor the original polynomial, we need to determine if this quadratic factor can be factored further. There are several methods for factoring quadratic expressions, including trial and error, using the quadratic formula, and factoring by grouping. In this case, we will use the factoring by grouping method. This method involves finding two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-1), which is -2, and add up to the middle coefficient (1). The numbers 2 and -1 satisfy these conditions, since 2 * -1 = -2 and 2 + (-1) = 1. Now, we rewrite the middle term (x) using these two numbers:
2x² + x - 1 = 2x² + 2x - x - 1
Next, we group the terms in pairs and factor out the greatest common factor (GCF) from each pair:
2x² + 2x - x - 1 = 2x(x + 1) - 1(x + 1)
Notice that both terms now have a common factor of (x + 1). We factor out this common factor:
2x(x + 1) - 1(x + 1) = (2x - 1)(x + 1)
Therefore, the quadratic factor 2x² + x - 1 factors into (2x - 1)(x + 1). Now we have the complete factorization of the original polynomial:
2x³ - 3x² - 3x + 2 = (x - 2)(2x² + x - 1) = (x - 2)(2x - 1)(x + 1)
This factorization expresses the polynomial as a product of three linear factors. Factoring the quadratic factor completes the process of factoring the original polynomial. This step is essential to obtain the complete set of factors and understand the polynomial's behavior fully. By factoring the quadratic, we can find the roots of the polynomial, which are the values of x that make the polynomial equal to zero. These roots correspond to the x-intercepts of the polynomial's graph.
Conclusion: The Complete Factorization and Its Significance
In conclusion, we have successfully factored the polynomial 2x³ - 3x² - 3x + 2 into its constituent factors. Starting with the knowledge that (x - 2) is a factor, we employed polynomial long division to find the other factor, which was the quadratic expression 2x² + x - 1. We then factored this quadratic expression by grouping, resulting in the factors (2x - 1) and (x + 1). Therefore, the complete factorization of the polynomial is:
2x³ - 3x² - 3x + 2 = (x - 2)(2x - 1)(x + 1)
This factorization reveals the roots of the polynomial, which are x = 2, x = 1/2, and x = -1. These roots are the values of x that make the polynomial equal to zero, and they correspond to the x-intercepts of the polynomial's graph. Understanding the complete factorization of a polynomial is crucial for various mathematical applications. It allows us to solve polynomial equations, simplify rational expressions, analyze the behavior of polynomial functions, and much more. Factoring is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts. In this guide, we have demonstrated a systematic approach to factoring polynomials, combining the Factor Theorem, polynomial long division, and factoring techniques for quadratic expressions. By practicing these techniques, you can develop proficiency in polynomial factorization and enhance your problem-solving abilities in mathematics.
