Name: polynomial division: quotient and remainder
Uploaded: 2025-11-26T05:43:58.000Z
Duration: 1 min 26 s
Channel: JoVE

Dividing polynomials using long division follows the same structure as numerical division.In this method, the polynomial being divided is the dividend, the one dividing it is the divisor, the result is the quotient, and any remaining part is the remainder.The division begins by writing the dividend and the divisor in standard form, with terms arranged in descending order of degree.The leading term of the dividend is divided by the leading term of the divisor to determine the first term of the quotient.This term is then multiplied by the entire divisor, and the result is subtracted from the dividend.After this, the next term in the dividend is brought down, and the process is repeated using the updated expression.These steps continue until the degree of the remaining expression is less than the degree of the divisor, where degree means the highest power of the variable.The final result can be expressed as the quotient plus the remainder over the divisor.For example, if a company&#8217;s total revenue and cost per item are modeled by polynomials, long division can be used to determine the expression indicating the revenue per item.

long division of polynomials

Polynomial division is an essential algebraic process to simplify expressions and solve equations. Just as numerical division separates a number into quotient and remainder, polynomial long division partitions a polynomial into simpler components; in this context, the dividend is the polynomial being divided, the divisor is the expression dividing it, and the result is expressed in terms of a quotient and a remainder.The division begins by arranging the dividend and divisor in standard form&#8212;terms in descending powers of the variable. The leading term of the dividend is divided by the divisor's leading term to determine the quotient's first term. This term is multiplied by the entire divisor, and the result is subtracted from the dividend. The next term in the dividend is brought down, and the process repeats until the remainder has a degree less than the divisor. The result is then expressed as:&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_1_1__editor_40055.svg' alt='40055_Equation_1_1_.svg' /></figure>For example, dividing:&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_2_1__editor_40055.svg' alt='40055_Equation_2_1_.svg' /></figure>the first term of the quotient is 6x, followed by &#8722;2, yielding the final quotient 6x - 2 and a remainder of 4. The result is written as:&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_3_1__editor_40055.svg' alt='40055_Equation_3_1_.svg' /></figure>This process can be generalized: for polynomials P(x) and D(x), there exist unique polynomials Q(x) and R(x) such that<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_4_2__editor_40055.svg' alt='40055_Equation_4_2_.svg' /></figure>where the degree of R(x) is less than that of D(x).

Long Division of Polynomials

Polynomial division is an essential algebraic process to simplify expressions and solve equations. Just as numerical division separates a number into ...

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Polynomial and Rational Functions

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polynomial division: quotient and remainder

Polynomial division breaks a polynomial into a quotient and a remainder. It works like numerical division, but the dividend is a polynomial and the divisor is an algebraic expression. The result shows how much of the divisor fits into the dividend and what is left over.

The first step is to write both polynomials in standard form, with terms in descending powers of the variable. Then divide the leading term of the dividend by the leading term of the divisor. That first result becomes the first term of the quotient.

Next, multiply that term by the entire divisor and subtract the result from the dividend. Bring down the next term and repeat the same steps. Continue until the remainder has a degree that is less than the degree of the divisor.

The final answer is written as a quotient plus a remainder. For example, dividing a polynomial can produce a quotient of 6x - 2 with a remainder of 4. In general, for polynomials P(x) and D(x), there are unique polynomials Q(x) and R(x) such that P(x) = D(x)Q(x) + R(x), and the degree of R(x) is less than the degree of D(x).

Polynomial Division: Quotient and Remainder

polynomial-division-quotient-and-remainder

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Polynomial division breaks a polynomial into a quotient and a remainder. It works like numerical division, but the dividend is a polynomial and the divisor is an algebraic expression. The result shows how much of the divisor fits into the dividend and what is left over. The first step is to write both polynomials in standard form, with terms in descending powers of the variable. Then divide the leading term of the dividend by the leading term of the divisor. That first result becomes the first...

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key terms and concepts

Learn polynomial long division by finding the quotient and remainder step by step. See how to use standard form, subtract, and stop when the remainder is smaller.

Polynomial Division: Quotient and Remainder | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Polynomial Division: Quotient and Remainder

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