Polynomial long division

Definition

Polynomial long division is a method used to divide one polynomial by another polynomial, similar to how long division is performed with numbers. This process involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by that result, and subtracting it from the dividend to find the remainder. This method helps simplify complex polynomial expressions and can reveal important factors and roots of polynomials.

5 Must Know Facts For Your Next Test

1. Polynomial long division can be used for both monomials and polynomials, allowing you to divide polynomials of varying degrees.
2. The process of polynomial long division requires organizing terms in descending order based on their degree to ensure clarity and accuracy during division.
3. If the degree of the dividend is less than that of the divisor, the result is simply the dividend with a remainder, indicating that the divisor cannot evenly divide into it.
4. After obtaining a quotient and a remainder from polynomial long division, you can express the result in the form: $$Q(x) + \frac{R(x)}{D(x)}$$ where $$Q(x)$$ is the quotient, $$R(x)$$ is the remainder, and $$D(x)$$ is the divisor.
5. Polynomial long division is particularly useful for simplifying rational expressions and solving polynomial equations, as it aids in finding factors and roots.

Review Questions

• How does polynomial long division compare to numerical long division in terms of steps and organization?
  • Polynomial long division follows a similar procedure to numerical long division but involves manipulating variables instead of numbers. In both cases, you divide, multiply, and subtract iteratively. The main difference lies in ensuring that polynomials are organized by their degrees in descending order, which keeps track of all terms correctly as you perform each step. This organization helps prevent errors when dealing with multiple variable terms.
• Explain how you would use polynomial long division to factor a polynomial expression. What role does this method play in finding factors?
  • To factor a polynomial using polynomial long division, you start by dividing the polynomial by one of its known factors or possible linear factors. By performing the division, you can identify the quotient and remainder. If the remainder is zero, then the divisor is indeed a factor of the original polynomial. This method allows you to break down complex polynomials into simpler factors, making it easier to solve equations or analyze polynomial behavior.
• Analyze a scenario where polynomial long division helps find the roots of a cubic polynomial. How does this process illustrate its practical application?
  • In a scenario where you have a cubic polynomial like $$P(x) = x^3 - 6x^2 + 11x - 6$$ and you suspect one root might be $$x = 1$$, you can use polynomial long division to divide $$P(x)$$ by $$x - 1$$. By performing this division, if there is no remainder, you confirm that $$x - 1$$ is a factor. The resulting quotient will be a quadratic polynomial, which can be solved using factoring or the quadratic formula to find all roots. This illustrates how polynomial long division not only simplifies expressions but also uncovers vital solutions within polynomials.

Related terms

Dividend: The polynomial that is being divided in a division operation.

Divisor:

The polynomial by which another polynomial (the dividend) is divided.

Remainder Theorem:

A theorem that states that if a polynomial is divided by a linear divisor, the remainder of that division is equal to the value of the polynomial at the root of the divisor.