Name: using synthetic division to find factors
Uploaded: 2025-11-26T05:43:56.000Z
Duration: 1 min 28 s
Channel: JoVE

综合除法是一种使用多项式系数进行多项式除法的简化方法。当除式为形如 x - c 的线性二项式时使用，其中 c 为实数。该过程首先将被除数的系数按照降幂顺序排列，从最高次项到常数项之间若有缺失的次数，需用零补齐。除数中的常数 c 放在除法结构的左侧。第一个系数写在横线下方。然后将其乘以常数，所得结果与下一个系数相加。这一乘加循环在整行中依次进行。行中的最后一个值表示余数，而前面的值构成商的系数——该商是一个次数比被除数低一次的多项式。最终结果可以表示为：被除数等于除数乘以商加上余数。在商业中，多项式可用于根据生产水平建模利润。为了计算特定生产水平下的利润，需将该多项式除以一个二项式 x 减去该生产水平。使用综合除法进行此项除法运算，所得余数即表示该水平下的利润。

多项式的综合除法

Synthetic Disvision of Polynomials

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Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Polynomial and Rational Functions

多项式与有理函数

KEY TERMS AND CONCEPTS

关键术语与概念

using synthetic division to find factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors.

The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table.

The process begins by bringing down the first coefficient unchanged. That number is then multiplied by c and added to the next coefficient. The same multiply-and-add pattern continues until every coefficient has been used.

The final row gives the quotient coefficients, except for the last number. That last number is the remainder. For example, dividing 2x^3 - 6x^2 + 2x - 4 by x - 2 gives the quotient 2x^2 - 2x - 2 and a remainder of zero.

A remainder of zero shows that x - 2 is a factor of the polynomial. This connects synthetic division to the Remainder Theorem, which says the remainder from dividing f(x) by x - c is f(c). If that remainder is zero, the Factor Theorem confirms that x - c is a factor of f(x).

Using Synthetic Division to Find Factors

quadratic models

Quadratic Motion and Parabolas

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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...

synthetic disvision of polynomials

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

real zeros of polynomials

Finding Polynomial X-Intercepts

Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis. The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the...

fundamental theorem of algebra

Polynomial Roots in Complex Numbers

Polynomial roots in complex numbers are a key result in algebra. The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex zero. For a polynomial of degree n, where n is at least 1 and the leading coefficient a n is not 0, there is at least one solution in the complex number system. This result also applies to polynomials with real coefficients, because the real numbers are a subset of the complex numbers. The Complete...

complex zeros

Finding Polynomial Roots With Complex Numbers

Complex numbers help describe every root of a polynomial equation. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = -1. A complex zero is any value that makes P(x) = 0 for a polynomial P(x). The complex number system includes the real numbers, so it gives a complete way to study polynomial roots. Every polynomial of degree n  1 can be written as a product of n linear factors over the complex numbers. In that factorization, each c_i...

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Rational functions are quotients of two polynomials, with the denominator Q(x) not equal to zero. Their graphs often include asymptotes, which are lines the graph gets close to but does not touch. These lines help show how the function behaves near certain input values and at large values of x. A vertical asymptote appears when the denominator is zero and the numerator is not zero. At those x-values, the function is undefined. To find a vertical asymptote, solve Q(x) = 0. For example, a...

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Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not. Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved,...

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Solving Nonlinear Inequalities on a Number Line

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key terms and concepts

Learn how synthetic division divides polynomials by x - c, finds the quotient and remainder, and shows when a binomial is a factor.

Using Synthetic Division to Find Factors | 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.

math fundamentals

Using Synthetic Division to Find Factors

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