Finding Coefficients In Polynomials Using Factor And Remainder Theorems

by ADMIN 72 views

In this article, we delve into the fascinating world of polynomial algebra, specifically focusing on how to determine unknown coefficients within a polynomial expression. We will explore the application of the Factor Theorem and the Remainder Theorem to solve a classic problem: Given a cubic polynomial with unknown coefficients, and provided that it has a specific factor and leaves a known remainder upon division by another linear expression, how can we find the values of these unknown coefficients?

Problem Statement

Let's consider the cubic polynomial:

P(x) = x³ + ax² + bx + 6

Our objective is to determine the values of the unknown coefficients, a and b, given the following two key pieces of information:

  1. (x - 2) is a factor of P(x): This implies that when we divide P(x) by (x - 2), the remainder is zero. In other words, P(2) = 0. This is a direct application of the Factor Theorem, which states that if (x - c) is a factor of a polynomial P(x), then P(c) = 0.

  2. P(x) leaves a remainder of 3 when divided by (x - 3): This tells us that P(3) = 3. This is a manifestation of the Remainder Theorem, which asserts that when a polynomial P(x) is divided by (x - c), the remainder is P(c).

Applying the Factor Theorem

Leveraging the Factor Theorem, we know that since (x - 2) is a factor of P(x), substituting x = 2 into the polynomial will yield zero. Let's perform this substitution:

P(2) = (2)³ + a(2)² + b(2) + 6 = 0

Simplifying this equation, we get:

8 + 4a + 2b + 6 = 0

Combining the constant terms, we have:

4a + 2b + 14 = 0

Dividing the entire equation by 2, we obtain our first linear equation in terms of a and b:

2a + b = -7

This equation establishes a crucial relationship between the unknown coefficients a and b. It signifies that any pair of values (a, b) that satisfies this equation will make (x - 2) a factor of the polynomial P(x).

Utilizing the Remainder Theorem

Now, let's employ the Remainder Theorem. We are given that P(x) leaves a remainder of 3 when divided by (x - 3). This means that when we substitute x = 3 into the polynomial, the result will be 3:

P(3) = (3)³ + a(3)² + b(3) + 6 = 3

Simplifying this equation, we get:

27 + 9a + 3b + 6 = 3

Combining the constant terms, we have:

9a + 3b + 33 = 3

Subtracting 33 from both sides, we get:

9a + 3b = -30

Dividing the entire equation by 3, we obtain our second linear equation in terms of a and b:

3a + b = -10

This equation provides us with another vital relationship between the coefficients a and b. It indicates that any pair of values (a, b) that satisfies this equation will result in a remainder of 3 when P(x) is divided by (x - 3).

Solving the System of Linear Equations

We now have a system of two linear equations with two unknowns:

  1. 2a + b = -7
  2. 3a + b = -10

To solve this system, we can use several methods, such as substitution, elimination, or matrix methods. For simplicity, let's use the elimination method. We can subtract the first equation from the second equation to eliminate the variable b:

(3a + b) - (2a + b) = -10 - (-7)

Simplifying, we get:

a = -3

Now that we have found the value of a, we can substitute it back into either of the original equations to solve for b. Let's substitute a = -3 into the first equation:

2(-3) + b = -7

Simplifying, we get:

-6 + b = -7

Adding 6 to both sides, we find:

b = -1

Therefore, the values of the unknown coefficients are a = -3 and b = -1.

Verification

To ensure the correctness of our solution, let's substitute the values of a and b back into the original polynomial and verify that the given conditions are satisfied:

P(x) = x³ - 3x² - x + 6

  1. Check if (x - 2) is a factor: We can do this by substituting x = 2 into P(x):

    P(2) = (2)³ - 3(2)² - (2) + 6 = 8 - 12 - 2 + 6 = 0

    Since P(2) = 0, (x - 2) is indeed a factor of P(x).

  2. Check the remainder when divided by (x - 3): We can do this by substituting x = 3 into P(x):

    P(3) = (3)³ - 3(3)² - (3) + 6 = 27 - 27 - 3 + 6 = 3

    Since P(3) = 3, the remainder when P(x) is divided by (x - 3) is indeed 3.

Thus, our solution is verified.

Conclusion

In conclusion, we have successfully determined the values of the unknown coefficients a and b in the cubic polynomial P(x) = x³ + ax² + bx + 6 by effectively utilizing the Factor Theorem and the Remainder Theorem. The Factor Theorem allowed us to establish a relationship between the coefficients based on the given factor (x - 2), while the Remainder Theorem provided another relationship based on the remainder when divided by (x - 3). By solving the resulting system of linear equations, we found that a = -3 and b = -1. We then verified our solution by substituting these values back into the original polynomial and confirming that it satisfied both given conditions. This problem exemplifies the power and elegance of polynomial algebra in solving real-world problems. Understanding and applying the Factor and Remainder Theorems are fundamental skills in algebra and are crucial for more advanced mathematical concepts. These theorems not only help in finding factors and remainders but also play a vital role in polynomial factorization, finding roots of polynomial equations, and analyzing polynomial behavior. Furthermore, the techniques used in this problem, such as solving systems of linear equations, are widely applicable in various fields of science, engineering, and economics.

In summary, the problem highlights the following key concepts:

  • Factor Theorem: If (x - c) is a factor of P(x), then P(c) = 0.
  • Remainder Theorem: When P(x) is divided by (x - c), the remainder is P(c).
  • Solving Systems of Linear Equations: Techniques like substitution and elimination are essential for finding unknown variables.
  • Verification: Always verify your solution to ensure accuracy.

By mastering these concepts, you can confidently tackle a wide range of polynomial problems and gain a deeper understanding of algebraic principles.