Polynomial Division Explained Dividing G(x) By (x - A)

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In the realm of algebra, polynomial division stands as a cornerstone technique for simplifying complex expressions and unraveling the relationships between polynomials. This article delves into the intricacies of polynomial division, focusing on the specific case of dividing the polynomial g(x) = 3x³ - 2x² + 7x - 1 by the linear factor (x - a). We'll meticulously dissect the long division process, mirroring John's method, and illuminate the underlying principles that govern this essential algebraic operation.

Polynomial long division, analogous to its arithmetic counterpart, serves as a systematic approach for dividing a polynomial (the dividend) by another polynomial (the divisor). The primary objective is to express the dividend as the product of the divisor and the quotient, along with any remainder. This can be succinctly represented as:

Dividend = (Divisor × Quotient) + Remainder

In our scenario, the dividend is g(x) = 3x³ - 2x² + 7x - 1, and the divisor is (x - a). Our mission is to determine the quotient and the remainder that result from this division.

Let's meticulously reconstruct John's long division process to gain a profound understanding of the mechanics involved:

  1. Initial Setup: Arrange the dividend and divisor in a long division format, ensuring that the terms are arranged in descending order of their exponents. We set up the problem as:
x - a | 3x³ - 2x² + 7x - 1
  1. First Iteration: Divide the leading term of the dividend (3x³) by the leading term of the divisor (x). This yields 3x², which becomes the first term of our quotient. Multiply the divisor (x - a) by 3x² to obtain 3x³ - 3ax². Subtract this result from the dividend:
     3x²
x - a | 3x³ - 2x² + 7x - 1
      -(3x³ - 3ax²)
      -----------------
           (3a - 2)x² + 7x

In this crucial step, we identify 3x² as the initial component of the quotient. Multiplying (x - a) by 3x² and subtracting the outcome from the dividend effectively eliminates the leading term (3x³), paving the way for subsequent iterations.

  1. Second Iteration: Divide the leading term of the new dividend ((3a - 2)x²) by the leading term of the divisor (x). This results in (3a - 2)x, which is added to the quotient. Multiply the divisor (x - a) by (3a - 2)x to get (3a - 2)x² - a(3a - 2)x. Subtract this from the current dividend:
     3x² + (3a - 2)x
x - a | 3x³ - 2x² + 7x - 1
      -(3x³ - 3ax²)
      -----------------
           (3a - 2)x² + 7x
           -((3a - 2)x² - a(3a - 2)x)
           --------------------------
                (7 + a(3a - 2))x - 1

This step mirrors the first iteration, focusing on the updated dividend. We determine the next term of the quotient, (3a - 2)x, and meticulously subtract the product of this term and the divisor from the current dividend.

  1. Third Iteration: Divide the leading term of the latest dividend ((7 + a(3a - 2))x) by the leading term of the divisor (x). This yields (7 + a(3a - 2)), which is appended to the quotient. Multiply the divisor (x - a) by (7 + a(3a - 2)) to obtain (7 + a(3a - 2))x - a(7 + a(3a - 2)). Subtract this from the current dividend:
     3x² + (3a - 2)x + (7 + a(3a - 2))
x - a | 3x³ - 2x² + 7x - 1
      -(3x³ - 3ax²)
      -----------------
           (3a - 2)x² + 7x
           -((3a - 2)x² - a(3a - 2)x)
           --------------------------
                (7 + a(3a - 2))x - 1
                -((7 + a(3a - 2))x - a(7 + a(3a - 2)))
                ----------------------------------
                     -1 + a(7 + a(3a - 2))

Continuing the iterative process, we identify the final term of the quotient, (7 + a(3a - 2)), and subtract the corresponding product from the dividend.

  1. Remainder: The final result of the subtraction, -1 + a(7 + a(3a - 2)), represents the remainder. As the degree of this expression is less than the degree of the divisor (x - a), we conclude the long division process.

From the long division process, we glean the following:

  • Quotient: 3x² + (3a - 2)x + (7 + a(3a - 2))
  • Remainder: -1 + a(7 + a(3a - 2))

Therefore, we can express g(x) as:

3x³ - 2x² + 7x - 1 = (x - a)(3x² + (3a - 2)x + (7 + a(3a - 2))) + (-1 + a(7 + a(3a - 2)))

The remainder we obtained, -1 + a(7 + a(3a - 2)), holds profound significance. According to the Remainder Theorem, when a polynomial g(x) is divided by (x - a), the remainder is equal to g(a). Let's verify this by evaluating g(a):

g(a) = 3a³ - 2a² + 7a - 1

Now, let's simplify our remainder expression:

-1 + a(7 + a(3a - 2)) = -1 + a(7 + 3a² - 2a) = -1 + 7a + 3a³ - 2a² = 3a³ - 2a² + 7a - 1

As anticipated, the remainder is indeed equal to g(a), reinforcing the Remainder Theorem's validity.

Polynomial division, exemplified by this comprehensive exploration, serves as a cornerstone in various mathematical contexts:

  • Factorization: If the remainder is zero, then (x - a) is a factor of g(x), facilitating polynomial factorization.
  • Root Finding: The roots of a polynomial, the values of x for which g(x) = 0, can be determined by identifying linear factors through division.
  • Simplification: Polynomial division can simplify complex expressions, making them more amenable to analysis and manipulation.
  • Calculus: In calculus, polynomial division finds application in integration and other operations.

Through the meticulous dissection of John's long division process, we have not only unveiled the quotient and remainder resulting from dividing g(x) = 3x³ - 2x² + 7x - 1 by (x - a) but also illuminated the fundamental principles underpinning polynomial division. The Remainder Theorem, a cornerstone of algebra, was validated through our calculations, underscoring the profound connections within the mathematical landscape. This exploration serves as a testament to the power and elegance of polynomial division, a technique that empowers us to unravel the intricacies of polynomial expressions and their underlying relationships.

Polynomial division, long division, remainder theorem, quotient, remainder, polynomial factorization, root finding, algebra, g(x) = 3x³ - 2x² + 7x - 1, (x - a)