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Polynomial operations encompass a range of algebraic techniques including addition, subtraction, multiplication, and division. This overview covers the basics of combining like terms, the FOIL method, polynomial composition, and the factoring of polynomials. It also delves into the role of polynomials in calculus, highlighting their ease of differentiation and integration, which are fundamental to mathematical analysis and problem-solving.

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## Addition and Subtraction of Polynomials

### Like Terms

Like terms are terms with identical variable factors that can be combined using the associative and commutative properties of addition

### Addition of Polynomials

Associative and Commutative Properties

The associative and commutative properties of addition govern the process of adding polynomials

Result of Addition

Adding polynomials results in a polynomial where like terms have been combined

### Subtraction of Polynomials

Taking the Difference of Like Terms

Subtraction of polynomials involves taking the difference of like terms, resulting in a polynomial where the corresponding coefficients have been subtracted

## Multiplication of Polynomials

### Distributive Property

Multiplication of polynomials requires the distributive property to multiply each term of one polynomial by every term of the other

### FOIL Method

The FOIL (First, Outer, Inner, Last) method is used to multiply binomials

### Result of Multiplication

Multiplying polynomials results in a polynomial where the degrees of the terms are the sums of the degrees of the terms being multiplied

## Composition of Polynomials

### Substitution

Composition of polynomials involves substituting one polynomial into another by replacing every occurrence of the variable in the first polynomial with the second polynomial

### Result of Composition

Composition of polynomials results in a new polynomial obtained by expanding and simplifying the expression

## Division and Rational Expressions

### Rational Expressions

Division of polynomials can result in rational expressions, which are ratios of polynomials analogous to fractions

### Polynomial Long Division and Synthetic Division

Division of polynomials can be performed using polynomial long division or synthetic division, resulting in a quotient and a remainder

### Factoring

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be used to simplify expressions, solve polynomial equations, and analyze polynomial functions

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