Learn Before
Dividing a Polynomial by a Binomial
To divide a polynomial by a binomial, use a procedure called polynomial long division, which closely mirrors the long division algorithm for whole numbers. Just as dividing 875 by 25 involves repeatedly dividing, multiplying, and subtracting digit groups, dividing a polynomial by a binomial involves the same cycle applied to polynomial terms.
The general procedure is:
- Set up the problem. Write the dividend (the polynomial being divided) under the long division bracket and the divisor (the binomial) outside. Make sure the dividend is in standard form — terms arranged in descending order of degree with no missing powers.
- Divide the leading term of the current dividend by the leading term of the divisor. Write the result in the quotient, positioned above the term of the dividend that matches its degree.
- Multiply the entire divisor by the term just written in the quotient. Align the resulting product under the corresponding terms of the dividend.
- Subtract. Change the signs of the product obtained in Step 3 and add (this is equivalent to subtracting). Then bring down the next term from the dividend.
- Repeat Steps 2–4 using the new expression obtained after subtraction and bringing down. Continue until the degree of the remaining expression is less than the degree of the divisor.
- Handle the remainder. If the remainder is zero, the division is exact. If a nonzero remainder results, express it as a fraction with the remainder in the numerator and the divisor in the denominator, and append it to the quotient.
To verify the result, multiply the quotient by the divisor — the product should equal the original dividend. When the divisor involves subtraction (for example, ), take extra care during the subtraction steps: it may be safer to change all signs and then add rather than subtracting directly, to avoid sign errors.
0
1
Tags
OpenStax
Elementary Algebra @ OpenStax
Ch.6 Polynomials - Elementary Algebra @ OpenStax
Algebra
Math
Prealgebra
Related
Finding the Quotient
Finding the Quotient
Finding the Quotient
Finding the Quotient
Dividing a Polynomial by a Binomial
In a business accounting spreadsheet, you are simplifying a formula where a multi-term cost expression (a polynomial) is divided by the number of departments (a monomial). Which of the following describes the correct algebraic step to perform this division?
A business analyst is simplifying a formula for 'Average Cost per Unit' where the total cost is a polynomial and the number of units is a monomial. Arrange the steps of the division process in the correct order according to the standard algebraic method.
In a corporate budget calculation, if you are dividing a polynomial (representing multiple expense categories) by a monomial (representing a single department), you only need to divide the first term of the polynomial by the monomial.
In a corporate finance department, analysts often decompose complex formulas to see individual cost drivers. Match each unified calculation (the single fraction) with its mathematically equivalent 'split' form. This is the essential first step in dividing a polynomial by a monomial.
Simplifying Multi-Term Budget Formulas
When a project manager is simplifying a formula that divides a polynomial (representing several different cost categories) by a monomial (representing a single time period), they must divide __________ term of the polynomial individually by the monomial before simplifying the result.
Logistics Cost Allocation Formula
Procedural Documentation for Formula Simplification
In a corporate finance spreadsheet, a 'Net Adjustment' formula divides a multi-term polynomial (representing various credit and debit accounts) by a negative monomial (representing a budget reduction factor). According to the algebraic rules for this operation, how should the signs of the individual terms in the resulting expression be determined?
In a professional data-modeling environment, once a complex multi-term expression has been split into individual fractions by a common divisor (a monomial), which two specific algebraic procedures are then applied to simplify each resulting individual fraction?
Finding the Quotient
Finding the Quotient
Finding the Quotient
Dividing a Polynomial by a Binomial Using Long Division
Learn After
Finding the Quotient
Finding the Quotient
Finding the Quotient
Inserting Placeholder Terms for Missing Degrees in Polynomial Division
Finding the Quotient
A data analyst is writing a technical manual for calculating resource distribution using polynomial division. Arrange the following steps of the polynomial long division process in the correct order for the manual.
A logistics analyst is using polynomial long division to model the distribution of inventory across multiple shipping zones. After completing the division steps, the analyst is left with a non-zero remainder. According to the standard procedure for polynomial division, how should this remainder be expressed in the final quotient?
A training coordinator is developing a cheat sheet for polynomial long division to help new analysts model resource distribution. Match each procedural term with its correct description based on the standard algorithm.
In a technical training manual for data analysts, the instructions for the 'Subtraction' step of polynomial long division state that an analyst should change the _______ of each term in the product and then add them to the current dividend to minimize the risk of mathematical errors.
Verifying Resource Distribution Models
True or False: In a technical model for resource distribution using polynomial long division, the standard procedure requires that the dividend be arranged in standard form, which involves listing terms in descending order of degree and including zero placeholders for any missing powers of the variable.
Technical SOP: Manual Verification of Polynomial Division
Auditing Algorithm Termination Logic
A technical manual for a mathematical processing engine describes the 'Multiplication' step of polynomial long division. According to the standard procedure, once a new term has been added to the quotient, it must be multiplied by which of the following?
In a professional Standard Operating Procedure (SOP) for manual mathematical auditing, the section on 'Step-by-Step Polynomial Division' specifies how to calculate each new term of the quotient. According to the standard algorithm, which specific components must be used during the 'Divide' step of each cycle?