For What Value Of M Is The Polynomial A(x) = X^2 + X - 1 A Divisor Of The Polynomial B(x) = X^4 + 4x^3 + X^2 + Mx + 1? Also, Find The Quotient.

by ADMIN 144 views

In the realm of mathematics, specifically algebra, polynomial division is a fundamental operation. It allows us to break down complex polynomial expressions into simpler, more manageable forms. This process is crucial for solving equations, simplifying expressions, and understanding the relationships between different polynomials. This comprehensive guide explores the concept of polynomial division, delving into its mechanics and applications. We will focus on a specific problem, determining the value of 'm' for which the polynomial A(x) = x^2 + x - 1 is a divisor of the polynomial B(x) = x^4 + 4x^3 + x^2 + mx + 1. This exploration will not only solidify your understanding of polynomial division but also highlight its importance in advanced algebraic manipulations. We will also find the quotient polynomial resulting from this division, further demonstrating the practical application of polynomial division techniques. Understanding polynomial division is essential for students, educators, and anyone working with mathematical models and equations. It lays the groundwork for more advanced topics in algebra and calculus, providing a solid foundation for mathematical reasoning and problem-solving. In this guide, we aim to provide a clear and concise explanation of the process, equipping you with the skills necessary to tackle a wide range of polynomial division problems.

Understanding Polynomial Divisibility

Polynomial divisibility is a key concept in algebra. A polynomial A(x) is said to be a divisor (or factor) of another polynomial B(x) if the division of B(x) by A(x) results in a quotient polynomial without any remainder. In simpler terms, if we can divide B(x) by A(x) and obtain another polynomial as the result, then A(x) is a divisor of B(x). This concept is analogous to integer divisibility; for example, 3 is a divisor of 12 because 12 divided by 3 equals 4, an integer. When dealing with polynomials, we look for a similar relationship. If B(x) can be written as A(x) multiplied by another polynomial Q(x), then A(x) is a divisor of B(x), and Q(x) is the quotient. The absence of a remainder is crucial in determining divisibility. A non-zero remainder indicates that A(x) is not a factor of B(x). This understanding of divisibility is fundamental to many algebraic operations, including factoring polynomials, solving equations, and simplifying expressions. For instance, if we know that (x - 2) is a factor of a polynomial, we can use this information to find the roots of the polynomial. Polynomial divisibility also plays a vital role in the study of polynomial functions and their graphs. The roots of a polynomial function correspond to the x-intercepts of its graph, and these roots can often be found by factoring the polynomial. In this guide, we will use the concept of polynomial divisibility to determine the value of 'm' for which A(x) is a divisor of B(x). This involves performing polynomial long division and ensuring that the remainder is zero. We will also explore the relationship between the coefficients of the polynomials and the conditions for divisibility.

Problem Statement: Finding 'm' and the Quotient

Let's clearly define the problem statement we aim to solve. We are given two polynomials:

  • A(x) = x^2 + x - 1
  • B(x) = x^4 + 4x^3 + x^2 + mx + 1

The core question is: For what value(s) of 'm' is the polynomial A(x) a divisor of the polynomial B(x)? In other words, we need to find the value(s) of 'm' such that when B(x) is divided by A(x), the remainder is zero. This condition ensures that A(x) is a factor of B(x). Furthermore, once we determine the value(s) of 'm', we need to find the quotient polynomial, Q(x), that results from the division. The quotient polynomial represents the other factor of B(x) when A(x) is a factor. This means B(x) can be expressed as A(x) * Q(x). Finding the quotient is an essential part of the polynomial division process. It provides additional insight into the relationship between the two polynomials. The process of finding 'm' and the quotient involves performing polynomial long division. We will divide B(x) by A(x), and the remainder will be an expression involving 'm'. By setting this remainder to zero, we can solve for 'm'. Once we have the value of 'm', we can substitute it back into B(x) and perform the division again to obtain the quotient Q(x). This problem combines algebraic manipulation, polynomial division, and equation-solving skills. It is a classic example of a problem that requires a thorough understanding of polynomial concepts.

Polynomial Long Division: A Step-by-Step Guide

The method of polynomial long division is the key to solving this problem. It's a systematic process for dividing one polynomial by another, much like the long division method used for dividing numbers. Let's break down the steps involved in polynomial long division:

  1. Set up the division: Write the dividend (B(x)) inside the division symbol and the divisor (A(x)) outside. Ensure that the polynomials are written in descending order of powers of x.
  2. Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient.
  3. Multiply the divisor by the quotient term: Multiply the entire divisor by the first term of the quotient.
  4. Subtract: Subtract the result from the dividend. This will give a new polynomial.
  5. Bring down the next term: Bring down the next term from the original dividend to the new polynomial.
  6. Repeat: Repeat steps 2-5 with the new polynomial until the degree of the remainder is less than the degree of the divisor.
  7. Remainder: The final polynomial obtained after the last subtraction is the remainder. If the remainder is zero, then the divisor divides the dividend evenly. In our specific problem, we will divide B(x) = x^4 + 4x^3 + x^2 + mx + 1 by A(x) = x^2 + x - 1. We will perform these steps carefully, keeping track of each term and its coefficient. The process will lead us to an expression for the remainder, which will be crucial for finding the value of 'm'. Polynomial long division is a skill that requires practice. It's essential to be meticulous and organized to avoid errors. Understanding the logic behind each step makes the process easier to remember and apply.

Performing the Division of B(x) by A(x)

Now, let's perform the polynomial long division with our specific polynomials:

B(x) = x^4 + 4x^3 + x^2 + mx + 1 and A(x) = x^2 + x - 1.

  1. Divide the leading term of B(x) (x^4) by the leading term of A(x) (x^2): x^4 / x^2 = x^2. This is the first term of our quotient.
  2. Multiply A(x) by x^2: x^2 * (x^2 + x - 1) = x^4 + x^3 - x^2
  3. Subtract this result from B(x): (x^4 + 4x^3 + x^2 + mx + 1) - (x^4 + x^3 - x^2) = 3x^3 + 2x^2 + mx + 1
  4. Now, divide the leading term of the new polynomial (3x^3) by the leading term of A(x) (x^2): 3x^3 / x^2 = 3x. This is the second term of our quotient.
  5. Multiply A(x) by 3x: 3x * (x^2 + x - 1) = 3x^3 + 3x^2 - 3x
  6. Subtract this result from the previous polynomial: (3x^3 + 2x^2 + mx + 1) - (3x^3 + 3x^2 - 3x) = -x^2 + (m + 3)x + 1
  7. Divide the leading term of the new polynomial (-x^2) by the leading term of A(x) (x^2): -x^2 / x^2 = -1. This is the third term of our quotient.
  8. Multiply A(x) by -1: -1 * (x^2 + x - 1) = -x^2 - x + 1
  9. Subtract this result from the previous polynomial: (-x^2 + (m + 3)x + 1) - (-x^2 - x + 1) = (m + 4)x

After performing the division, we have a remainder of (m + 4)x. The quotient is x^2 + 3x - 1. The remainder is the key to finding the value of 'm'. In the next section, we will analyze this remainder and determine the condition for A(x) to be a divisor of B(x).

Determining the Value of 'm'

Recall that for A(x) to be a divisor of B(x), the remainder from the polynomial long division must be zero. In the previous section, we found that the remainder is (m + 4)x. For this remainder to be zero for all values of x, the coefficient of x must be zero. Therefore, we must have:

m + 4 = 0

Solving this equation for 'm', we get:

m = -4

This is the value of 'm' for which A(x) is a divisor of B(x). When m = -4, the polynomial B(x) becomes:

B(x) = x^4 + 4x^3 + x^2 - 4x + 1

Now that we have the value of 'm', we can verify our result by substituting it back into the division and checking if the remainder is indeed zero. This step is crucial to ensure that we have not made any errors in our calculations. The value m = -4 is a specific solution that satisfies the divisibility condition. It allows us to express B(x) as a product of A(x) and the quotient polynomial, which we will discuss in the next section. Finding the value of 'm' is a critical step in many polynomial problems, especially those involving factorization and solving equations.

Finding the Quotient Polynomial

We've already found the quotient polynomial during the long division process. From the steps outlined earlier, the quotient Q(x) is:

Q(x) = x^2 + 3x - 1

This means that when m = -4, B(x) can be expressed as the product of A(x) and Q(x):

B(x) = A(x) * Q(x)

Substituting the polynomials, we have:

x^4 + 4x^3 + x^2 - 4x + 1 = (x^2 + x - 1) * (x^2 + 3x - 1)

This result confirms that A(x) is indeed a divisor of B(x) when m = -4. The quotient polynomial provides valuable information about the relationship between A(x) and B(x). It tells us how many times A(x)