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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Like terms are terms with identical variable factors that can be combined using the associative and commutative properties of addition
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
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 requires the distributive property to multiply each term of one polynomial by every term of the other
The FOIL (First, Outer, Inner, Last) method is used to multiply binomials
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 involves substituting one polynomial into another by replacing every occurrence of the variable in the first polynomial with the second polynomial
Composition of polynomials results in a new polynomial obtained by expanding and simplifying the expression
Division of polynomials can result in rational expressions, which are ratios of polynomials analogous to fractions
Division of polynomials can be performed using polynomial long division or synthetic division, resulting in a quotient and a remainder
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
00
______ are expressions with variables and coefficients, using operations like addition and multiplication, and variables raised to ______ exponents.
Polynomials
non-negative integer
01
FOIL Method Application
FOIL used for binomials: multiply First, Outer, Inner, Last terms, then combine like terms.
02
Polynomial Multiplication Closure
Multiplying polynomials always yields another polynomial; operation is closed.
