Division of a polynomial by another polynomial | Part-4

Division of a Polynomial by Another Polynomial: Finding Unknowns and Real-World Applications

Dividing one polynomial by another is a foundational concept in algebra, particularly useful when the divisor is a known factor of the dividend. This method not only simplifies expressions but also helps determine unknown values and solve real-world problems efficiently.

Understanding Polynomial Division

Polynomial division involves dividing a higher-degree polynomial (called the dividend) by a lower-degree polynomial (called the divisor). The result is a quotient and, in some cases, a remainder. When the divisor is a factor of the dividend, the remainder is zero, which has important implications in solving equations and factorization.

Example: Dividing Polynomials and Finding an Unknown

Let’s take an example of a polynomial division problem:

  (2x³ + 3x² - x - 6) ÷ (x + 2)
  

Let’s use polynomial long division:

1. Divide the first term: 2x³ ÷ x = 2x²
2. Multiply: 2x² × (x + 2) = 2x³ + 4x²
3. Subtract: (2x³ + 3x²) - (2x³ + 4x²) = -x²
4. Bring down the next term: -x
5. Divide: -x² ÷ x = -x
6. Multiply: -x × (x + 2) = -x² - 2x
7. Subtract: (-x² - x) - (-x² - 2x) = x
8. Bring down: -6
9. Divide: x ÷ x = 1
10. Multiply: 1 × (x + 2) = x + 2
11. Subtract: (x - 6) - (x + 2) = -8

Quotient: 2x² - x + 1

Remainder: -8

Finding an Unknown When Divisor Is a Factor

If we are told that (x + 2) is a factor of the dividend, then the remainder must be zero. Suppose the polynomial is:

  2x³ + 3x² + ax - 6
  

And we know that (x + 2) is a known factor. To find the unknown a, we apply polynomial division and ensure that the remainder becomes zero.

After performing the division as before (with a replacing the -1 term), set the remainder to zero and solve for a. Alternatively, use the **Remainder Theorem**, which says:

For (x + 2) to be a factor, the polynomial must satisfy f(-2) = 0.
  

Substitute -2 into the polynomial:

  f(x) = 2x³ + 3x² + ax - 6
  f(-2) = 2(-8) + 3(4) + a(-2) - 6 = -16 + 12 - 2a - 6 = -10 - 2a
  

Set equal to zero:

  -10 - 2a = 0 → a = -5
  

Answer: a = -5 makes the divisor a factor of the dividend.

Applications of Algebraic Division

Algebraic division has several important real-life and academic applications, including:

• Solving equations: When polynomials are set equal to zero, factoring helps find the roots or solutions.
• Simplifying expressions: Division reduces complex expressions to simpler forms, useful in calculus and higher math.
• Checking for factors: Verifying whether one polynomial is a factor of another helps in polynomial factorization.
• Error detection in coding: Polynomial division is used in algorithms for CRC (Cyclic Redundancy Check) in digital communication.

Dividing one polynomial by another reveals not just the quotient and remainder—it can also help determine unknown coefficients and verify factor relationships. When the divisor is a factor of the dividend, division becomes a powerful tool for solving algebraic expressions and real-world problems. With a solid understanding of polynomial division and the ability to apply concepts like the Remainder Theorem, students can confidently approach a wide range of mathematical challenges.