Lecture 4: Polynomials : Lecture 4: Polynomials Josh C. Shott
Objectives : Objectives Define a polynomial.
Classify a polynomial by degree.
Add, subtract, and multiply polynomials. Algebra Lecture 3 by Josh C. Shott
Polynomials : Polynomials An arithmetic expression consisting of known values and unknown values. Algebra Lecture 4 by Josh C. Shott variable – an arbitrary value in an expression. The value of this expression is unknown, and generally represented with a letter. Variables ARE allowed to change or vary, hence the name variable. algebraic expressions constant – a fixed value in an expression. The value of this expression is known, and more importantly is NOT allowed to change Examples term – constants and variables multiplied together. constant variable In a term, the constant is referred to as a coefficient.
Polynomials : Polynomials For any real number x and natural number n, then the expression xn is defined as the number x multiplied by itself n times. Algebra Lecture 4 by Josh C. Shott exponents Examples base – the number that is being multiplied by itself (x) base exponent exponent – the number of times the base is being multiplied by itself (n)
Polynomials : Polynomials Any base raised to the zero power (other than zero) is equal to 1. Algebra Lecture 4 by Josh C. Shott exponents Examples
Polynomials : When a variable has a negative exponent. (variables are in the denominator):
Variables have rational exponents that do not simplify to a natural number. (Variables are under a radicand):
Variables are raised to an irrational power: Polynomials A one term expression where each variable is only raised to a natural number power or zero. Algebra Lecture 4 by Josh C. Shott monomial Terms that are NOT Monomials:
Polynomials : Polynomials The sum or difference of two monomials. Algebra Lecture 4 by Josh C. Shott binomial polynomial The sum two or more monomials. (Note the coefficient of the monomial can be negative.) trinomial A polynomial with three terms. Examples
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Explain why the following are NOT polynomials: x is in the denominator y is raised to the -3 power y is in the denominator y is raised to the ½ power x is in the radicand
Polynomials : Polynomials The degree of a monomial is the value of the exponent in the term. Algebra Lecture 4 by Josh C. Shott degree of a one variable monomial degree of a polynomial The degree of a polynomial is the same as the degree of the highest term in the polynomial degree of a two or more variable monomial The degree of a monomial with more than one variable is the sum of the exponents in the term. x2 has degree 2.
3x5 has degree 5.
3 has degree 0. xy has degree 2.
3x5y2 has degree 7.
4stuvyz has degree 6. x2 – x - 5 has degree 2.
3xy – x +y has degree 2.
3xyz – x2y2 has degree 4.
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Determine the degree of each polynomial 2 2 4 0 0 3 4 10
Polynomial Operations : Polynomial Operations To add two polynomials, remove parenthesis and combine like terms. Algebra Lecture 4 by Josh C. Shott addition subtraction To subtract one polynomial from another, change the sign of every term in the polynomial you are subtracting, and then add them by combining like terms: vertical alignment Sometimes it helps to vertically align the terms in the polynomial:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following sums and differences:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following sums and differences:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following sums and differences:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following sums and differences:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following sums and differences:
Polynomial Operations : Polynomial Operations Algebra Lecture 4 by Josh C. Shott multiplication When multiplying exponents of the same base, we add the exponents distributive property When multiplying a monomial by a polynomial, we use the distributive property:
Polynomial Operations : Polynomial Operations Algebra Lecture 4 by Josh C. Shott Multiplying a polynomial by a polynomial, is similar to arithmetic. You do this every time you multiply 2 digit numbers without knowing it. Example: is actually
Polynomial Operations : Polynomial Operations Algebra Lecture 4 by Josh C. Shott vertical multiplication
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following products:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following products:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following products:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Find the following products:
A Shortcut to Multiplication : A Shortcut to Multiplication Algebra Lecture 4 by Josh C. Shott When multiplying two binomials: First Outer Inner Last
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Perform the following multiplications:
Practice Examples : Practice Examples Algebra Lecture 4 by Josh C. Shott Perform the following multiplications:
Slide 27 : ? Thank you for your time Algebra Lecture 4 by Josh C. Shott