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HomeHomework HelpmathematicsPolynomial Division

Polynomial Division

Polynomial division involves determining how many times one polynomial can be divided by another, while row reduction is a method for simplifying matrices to solve systems of linear equations.

intermediate
3 hours
Mathematics
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Overview

Polynomial division and row reduction are fundamental concepts in algebra that help simplify complex expressions and solve equations. Polynomial division allows us to break down polynomials into simpler parts, making it easier to analyze their behavior and find roots. Row reduction, on the other han...

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Key Terms

Polynomial
An expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

Example: 3x² + 2x - 5

Quotient
The result of division, representing how many times one number is contained within another.

Example: In 10 ÷ 2 = 5, 5 is the quotient.

Remainder
The amount left over after division when one number does not divide evenly into another.

Example: In 10 ÷ 3, the remainder is 1.

Matrix
A rectangular array of numbers arranged in rows and columns.

Example: [[1, 2], [3, 4]]

Row Echelon Form
A form of a matrix where all non-zero rows are above any rows of all zeros, and leading coefficients are to the right of the leading coefficients in the rows above.

Example: A matrix in row echelon form might look like [[1, 2, 3], [0, 1, 4]].

Reduced Row Echelon Form
A form of a matrix where it is in row echelon form and all leading coefficients are 1 and are the only non-zero entries in their columns.

Example: A matrix in reduced row echelon form might look like [[1, 0, 0], [0, 1, 0]].

Related Topics

Factoring Polynomials
Understanding how to factor polynomials is essential for simplifying expressions and solving equations.
intermediate
Linear Algebra
Linear algebra involves the study of vectors, matrices, and linear transformations, which are foundational in many areas of mathematics.
advanced
Systems of Equations
Solving systems of equations is a key application of both polynomial division and row reduction techniques.
intermediate

Key Concepts

Long DivisionSynthetic DivisionRow Echelon FormReduced Row Echelon Form