Which Factor Of X³ - 5x² - 2x + 24 Is Given If X - 3 Is One Of The Linear Factors?
Factoring polynomials is a fundamental skill in algebra, essential for solving equations, simplifying expressions, and understanding the behavior of functions. In this comprehensive guide, we'll delve into the process of factoring the polynomial x³ - 5x² - 2x + 24, given that (x - 3) is one of its linear factors. This article will not only provide the solution but also equip you with the knowledge and techniques to tackle similar problems with confidence. We'll explore the factor theorem, polynomial division, and other factoring methods, ensuring a thorough understanding of the concepts involved. Let's embark on this journey to master polynomial factorization.
Understanding the Problem
Before diving into the solution, let's clearly define the problem. We are given the cubic polynomial x³ - 5x² - 2x + 24 and told that (x - 3) is one of its factors. Our goal is to find another factor from the following options:
- x + 4
- x - 2
- x + 2
- x² - 8x + 22
To solve this, we will use the factor theorem and polynomial division. The factor theorem states that if (x - a) is a factor of a polynomial P(x), then P(a) = 0. Conversely, if P(a) = 0, then (x - a) is a factor of P(x). Polynomial division allows us to divide the given polynomial by the known factor (x - 3) to find the remaining factor.
Applying the Factor Theorem
Since we know that (x - 3) is a factor, we can verify this using the factor theorem. Let P(x) = x³ - 5x² - 2x + 24. If (x - 3) is a factor, then P(3) should equal 0. Let's substitute x = 3 into the polynomial:
P(3) = (3)³ - 5(3)² - 2(3) + 24 P(3) = 27 - 5(9) - 6 + 24 P(3) = 27 - 45 - 6 + 24 P(3) = 0
As P(3) = 0, this confirms that (x - 3) is indeed a factor of the polynomial. Now, we can proceed with polynomial division to find the other factor.
Polynomial Long Division
Polynomial long division is a method used to divide a polynomial by another polynomial of a lower or equal degree. In our case, we will divide x³ - 5x² - 2x + 24 by (x - 3). Here's how the long division process works:
- Set up the division:
________________________
x - 3 | x³ - 5x² - 2x + 24
-
Divide the first term of the dividend (x³) by the first term of the divisor (x):
x³ / x = x²
Write x² above the x² term in the dividend.
x² ____________________
x - 3 | x³ - 5x² - 2x + 24
-
Multiply the divisor (x - 3) by the quotient term (x²):
x²(x - 3) = x³ - 3x²
Write the result below the corresponding terms in the dividend.
x² ____________________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
- Subtract the result from the dividend:
x² ____________________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
---------
-2x² - 2x
- Bring down the next term from the dividend (-2x):
x² ____________________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
---------
-2x² - 2x
-
Divide the first term of the new dividend (-2x²) by the first term of the divisor (x):
-2x² / x = -2x
Write -2x next to x² in the quotient.
x² - 2x ______________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
---------
-2x² - 2x
-
Multiply the divisor (x - 3) by the new quotient term (-2x):
-2x(x - 3) = -2x² + 6x
Write the result below the corresponding terms in the new dividend.
x² - 2x ______________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
---------
-2x² - 2x
-2x² + 6x
- Subtract the result from the new dividend:
x² - 2x ______________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
---------
-2x² - 2x
-2x² + 6x
---------
-8x + 24
- Bring down the next term from the dividend (+24):
x² - 2x ______________
x - 3 | x³ - 5x² - 2x + 24
x³ - 3x²
---------
-2x² - 2x
-2x² + 6x
---------
-8x + 24
-
Divide the first term of the new dividend (-8x) by the first term of the divisor (x):
-8x / x = -8
Write -8 next to -2x in the quotient.
x² - 2x - 8 ________ x - 3 | x³ - 5x² - 2x + 24 x³ - 3x² --------- -2x² - 2x -2x² + 6x --------- -8x + 24 -
Multiply the divisor (x - 3) by the new quotient term (-8):
-8(x - 3) = -8x + 24
Write the result below the corresponding terms in the new dividend.
x² - 2x - 8 ________ x - 3 | x³ - 5x² - 2x + 24 x³ - 3x² --------- -2x² - 2x -2x² + 6x --------- -8x + 24 -8x + 24 -
Subtract the result from the new dividend:
x² - 2x - 8 ________ x - 3 | x³ - 5x² - 2x + 24 x³ - 3x² --------- -2x² - 2x -2x² + 6x --------- -8x + 24 -8x + 24 --------- 0
The remainder is 0, which confirms that (x - 3) is a factor. The quotient is x² - 2x - 8, which is the other factor we were looking for.
Factoring the Quadratic Factor
Now we have the quadratic factor x² - 2x - 8. To find its linear factors, we can factor it further. We are looking for two numbers that multiply to -8 and add to -2. These numbers are -4 and 2.
Therefore, we can factor the quadratic as follows:
x² - 2x - 8 = (x - 4)(x + 2)
So, the complete factorization of the polynomial is:
x³ - 5x² - 2x + 24 = (x - 3)(x - 4)(x + 2)
Identifying the Correct Answer
From the given options, we need to identify which one is a factor of the polynomial. We found that the factors are (x - 3), (x - 4), and (x + 2). Comparing these with the options:
- x + 4
- x - 2
- x + 2
- x² - 8x + 22
The correct answer is (x + 2).
Conclusion
In this comprehensive guide, we successfully factored the polynomial x³ - 5x² - 2x + 24, given that (x - 3) is one of its linear factors. We utilized the factor theorem to verify that (x - 3) is indeed a factor and then employed polynomial long division to find the remaining quadratic factor, x² - 2x - 8. Finally, we factored the quadratic factor into (x - 4)(x + 2). Therefore, the factors of the polynomial are (x - 3), (x - 4), and (x + 2). From the given options, (x + 2) is the correct answer. This exercise demonstrates the power and elegance of algebraic techniques in solving polynomial factorization problems. Mastering these techniques is crucial for further studies in mathematics and related fields.
Key Takeaways:
- The factor theorem is a powerful tool for verifying factors of a polynomial.
- Polynomial long division helps in dividing polynomials and finding the remaining factors.
- Factoring quadratic expressions involves finding two numbers that multiply to the constant term and add up to the coefficient of the linear term.
- Understanding these concepts and techniques is essential for solving a wide range of algebraic problems.
This detailed explanation provides not only the solution but also a thorough understanding of the underlying concepts and methods. By following this guide, you can confidently approach similar polynomial factorization problems and enhance your algebraic skills.