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Remainder Theorem
The Remainder Theorem states that if a polynomial function is divided by a binomial of the form , then the remainder of that division is exactly equal to the value of the function evaluated at , which is . This theorem can be logically derived by expressing the division in function notation: to get the dividend , multiply the quotient by the divisor , and add the remainder . This gives the equation . If this equation is evaluated at , it yields , which simplifies to , leaving .
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Intermediate Algebra @ OpenStax
Ch.5 Polynomials and Polynomial Functions - Intermediate Algebra @ OpenStax
Algebra
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Remainder Theorem
Polynomial Division Check in Function Notation
Imagine your company uses a polynomial function, , to model projected weekly profit based on , the number of products manufactured. Your manager asks you to evaluate this profit function for an input of 500 products. Which sequence of steps correctly describes the mathematical procedure you must follow?
A facility manager uses a mathematical expression to estimate the weight capacity of a storage floor based on its thickness. To find the capacity for a specific thickness, the manager must evaluate the expression. Arrange the steps of this evaluation process in the correct sequence.
Procedural Steps for Evaluating Models
An operations manager at a logistics company uses polynomial functions to predict fuel consumption based on vehicle weight. To analyze these models correctly, the manager must understand the specific components of the mathematical process. Match each term with the description of its role in this analysis.
A logistics company uses a polynomial function to model fuel consumption based on vehicle speed. True or False: To evaluate this function for a specific speed, that value must be substituted for the variable in every term of the polynomial expression before the order of operations is used to simplify the result.
Learn After
Factor Theorem
Finding the Remainder Using the Remainder Theorem for f(x) = x^3 + 3x + 19 Divided by x + 2
Finding the Remainder Using the Remainder Theorem for f(x) = x^3 + 4x + 15 Divided by x + 2
Finding the Remainder Using the Remainder Theorem for f(x) = x^3 - 7x + 12 Divided by x + 3
A corporate analyst uses a polynomial function to model the quarterly revenue of a regional branch. To check the accuracy of their model for the 10th quarter, they divide the polynomial by the expression . According to the Remainder Theorem, which value is equivalent to the remainder of this division?
You are an analyst reviewing a report that models monthly expenses using a polynomial . According to the report's explanation of the Remainder Theorem, to find the remainder when dividing the polynomial by , you should evaluate the function at .
A production manager uses a polynomial function to model the weekly operating costs of a manufacturing plant. To find the remainder when dividing by various divisors using the Remainder Theorem, match each divisor with the correct functional evaluation that represents that remainder.
Verifying Polynomial Remainders in Financial Models
A technical writer is documenting the formal derivation of the Remainder Theorem for an algebra training manual. Arrange the following steps in the correct logical order to demonstrate why the remainder of a polynomial divided by is equal to .