5.4 Dividing Polynomials
2 min read•june 24, 2024
Dividing polynomials is a crucial skill in algebra. It involves breaking down complex expressions into simpler parts, making it easier to solve equations and understand relationships between variables. This process is essential for tackling real-world problems in fields like engineering and physics.
has practical applications in areas like geometry and physics. By mastering techniques like and , you'll be able to solve problems involving area, volume, and other mathematical relationships more efficiently.
Dividing Polynomials
Polynomial division methods
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- of polynomials
- Divide leading term of by leading term of
- Multiply by and subtract result from
- Repeat process until of is less than of divisor
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- Arrange coefficients of dividend in descending order of degree
- Write negative of constant term of divisor outside division bracket
- Multiply first coefficient by constant term and add result to second coefficient
- Repeat process until all coefficients have been used
- Last number in division bracket is , other numbers form
Applications of polynomial division
- Area problems
- Divide polynomial representing area by polynomial representing one side to find other side
- Rectangle with area and one side , divide area polynomial by to find other side
- Volume problems
- Divide polynomial representing volume by polynomial representing area of base to find height
- Cylinder with volume and base area , divide volume polynomial by to find height
Factors using remainder theorem
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- Polynomial divided by , remainder equals
- If , then is factor of
- Finding factors using remainder theorem
- Evaluate polynomial at various values of to find remainders
- If remainder is 0, corresponding is factor of polynomial
- To find factors of , evaluate polynomial at
- If , then is factor
- If , then is factor
- If , then is factor
Polynomial Characteristics and Operations
- Degree: The highest power of the variable in a polynomial
- : The coefficient of the term with the highest degree
- : An expression that consists of variables and coefficients combined using addition, subtraction, and multiplication
- : The process of breaking down a polynomial into the product of simpler polynomials
- : A fraction where both the numerator and denominator are polynomials
Key Terms to Review (31)
÷: The division symbol, also known as the obelus, is a mathematical operator used to indicate the division of one number by another. It represents the process of splitting a quantity into equal parts or determining how many times one number is contained within another.
Cubic Polynomial: A cubic polynomial is a polynomial of degree three, meaning it contains a term with the variable raised to the power of three. Cubic polynomials are an important class of functions in algebra and have unique properties that distinguish them from linear and quadratic polynomials.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
Dividend: A dividend is the polynomial that is being divided by another polynomial (the divisor) in a division problem. It is the numerator in the fraction form of polynomial division.
Dividend: A dividend is the portion of a company's profits that is distributed to its shareholders. It represents a return on the shareholders' investment in the company's stock.
Divisor: A divisor is a polynomial that divides another polynomial, known as the dividend, without leaving a remainder. In algebraic terms, if $f(x)$ and $g(x)$ are polynomials, then $g(x)$ is a divisor of $f(x)$ if $f(x) = g(x) \cdot q(x) + r(x)$ and $r(x) = 0$.
Divisor: A divisor is a factor that divides another number or expression without leaving a remainder. It is a fundamental concept in the operation of division, where the divisor is the number by which another number or expression is divided.
Factor Theorem: The Factor Theorem is a fundamental principle in polynomial algebra that establishes a relationship between the factors of a polynomial and its zeros. It provides a way to determine whether a given expression is a factor of a polynomial and to find the roots or zeros of a polynomial function.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Long division: Long division is a method used to divide polynomials by another polynomial of lesser or equal degree. It involves repeated division, multiplication, and subtraction to obtain the quotient and remainder.
Long Division: Long division is a step-by-step procedure for dividing one polynomial by another, where the divisor is of a higher degree than the dividend. It involves repeatedly subtracting multiples of the divisor from the dividend until the remainder is of a lower degree than the divisor.
Monic Polynomial: A monic polynomial is a polynomial where the leading coefficient, the coefficient of the highest degree term, is equal to 1. This means that the highest degree term in the polynomial has a coefficient of 1, while the other coefficients can be any real numbers.
Polynomial Division: Polynomial division is the process of dividing one polynomial by another to find the quotient and remainder. It is a fundamental operation in algebra that allows for the factorization and simplification of polynomial expressions.
Polynomial function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can be expressed in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $a_i$ are constants and $n$ is a non-negative integer.
Polynomial Function: A polynomial function is an algebraic function that can be expressed as the sum of a finite number of non-negative integer powers of a variable, with coefficients. Polynomial functions are a fundamental concept in algebra and are closely related to topics such as power functions, polynomial division, and the zeros of polynomial functions.
Polynomial Long Division: Polynomial long division is a method used to divide one polynomial by another polynomial. It involves a step-by-step process of dividing the terms of the dividend by the terms of the divisor, similar to the long division algorithm used for dividing integers.
Quotient: The quotient is the result obtained when one polynomial is divided by another. It represents the factor by which the divisor multiplies to yield the dividend.
Quotient: The quotient is the result of dividing one number or expression by another. It represents the number of times the divisor goes into the dividend, or the amount that is left over from the division process.
R(x): R(x) is a mathematical function that represents the remainder of a polynomial division operation. It is a key concept in the topic of dividing polynomials, as it allows for the determination of the quotient and remainder when one polynomial is divided by another.
Rational expression: A rational expression is a fraction where both the numerator and the denominator are polynomials. The denominator cannot be zero.
Rational Expression: A rational expression is a mathematical expression that consists of one or more polynomials divided by one or more polynomials. It represents a ratio of two polynomial functions and can be used to model and analyze various mathematical relationships.
Rational Root Theorem: The Rational Root Theorem is a fundamental principle in the study of polynomial functions that provides a way to determine the possible rational roots of a polynomial equation. It helps simplify the process of finding the roots or zeros of a polynomial by narrowing down the potential solutions.
Remainder: The remainder is the part of a polynomial that is left over after division by another polynomial. It can be found using techniques such as polynomial long division or synthetic division.
Remainder: The remainder is the amount left over when one number is divided by another. It represents the portion of the dividend that is not evenly divisible by the divisor, and it is the value that remains after the division process has been completed.
Remainder Theorem: The Remainder Theorem is a fundamental principle in polynomial division that states the relationship between the division of a polynomial by a linear expression and the value of the polynomial when the variable is set to a specific value. It provides a way to determine the remainder of a polynomial division without actually performing the long division process.
Synthetic division: Synthetic division is a simplified method of dividing polynomials where only the coefficients are used. It is particularly useful for dividing by linear factors of the form $x - c$.
Synthetic Division: Synthetic division is a shortcut method used to divide a polynomial by a linear expression of the form $(x - a)$. It allows for the efficient computation of polynomial division, providing a streamlined approach to determining the quotient and remainder of the division process.