MTH95 Day 6 Sections 5.3 & 7.1 Section 5.3 Polynomials and Polynomial Functions Definitions: Term Constant Factor Coefficient Polynomial Monomial Binomial Trinomial Degree of a term Degree of a Polynomial Now here is something that is symbolic: 3x 4 4x 2 + x 7 This is a polynomial, and it is also known as a polynomial in one variable or a polynomial in x. Most of the polynomials what we will be working with in this class are in one variable, so let s just call this a polynomial. Within this polynomial there are 4 terms. A term is a number or variable or the product of a number and variables. 3x 4 is a term, consisting of the factors 3 and a bunch of x s. Factors are things connected by multiplication, while terms are separated from each other by addition or subtraction. Can you identify the 4 terms in this polynomial? Notice that we are switching from the subtraction to adding the negative of the following term For the term 3x 4, the 3 is called the coefficient. For 4x 2, 4 is the coefficient. What about the term x? That has an understood coefficient of 1, x = 1x. The final term in the polynomial is 7, and you may have noticed that this term does not have any variables, so this term is called a constant term constant for short. Do you know what mono, bi, tri and poly mean? Monomial has one term: 3x 2 4 x 6 Binomial has two terms: x + 2 x 2 4 72x 6 81 Trinomial has three terms: 2x 2 4x + 8 And all of these can still be called polynomials. OK, time to actually define a polynomial: A polynomial has one or more terms, where the coefficients of each term are real numbers and the exponents of any variables are non negative integers. NOT polynomials 3x 2 7x+2 Negative Exponent Negative Exponent Non real coefficient 1
By the way, have you noticed how I write the terms so that the exponents for the terms are in descending order? That is a good habit to maintain. Speaking of the exponents, let s talk about degree. If a polynomial only has one type of variable, the degree of a term is the value of the exponent for that term. 2x 5 + 3x 2 x + 7 Degree of each term: The degree of the entire polynomial is the degree of whichever term has the highest degree. Which is why we like to write polynomials in descending order, the degree of the polynomial is the degree of the first term, and by the way, the coefficient of the term with the highest degree is called the leading coefficient. Here is the weird thing. Say you have a polynomial that is not in one variable. Something like: 3x 4 y 3 + 4x 2 y 3 7xy + 8 Then to get the degree of each term you add up the exponents of the variables in the term. For the term 3x 4 y 3, the degree is 7 Polynomials are expressions, not equations. So we cannot solve a polynomial unless we set it equal to something like another polynomial. They are functions, so we can name them: P x = x 2 + 5x 6 and evaluate them P 2 = 2 2 + 5 2 6 = 4 + 10 6 = 0. We can simplify, add, and subtract polynomials by combining like terms: 3x 2 7x + 1 + 7x 2 + 8x 2 3a 2 +8a 7 5a 3 2
Section 7.1 Radicals and Radical Functions Let s talk about inverse functions for a second. Subtracting undoes adding, dividing undoes multiplying. Wha undoes squaring undoes an exponent of 2? The square root function. The radicand is inside the radical: for a, a nonnegative number. These have all been nice rational numbers, because they were perfect squares. The square root of a number that is not a perfect square will be an irrational number. Which is still a real number, it can be located on a number line. But irrational numbers can be difficult to write because they have non repeating, non terminating decimals, and we usually end up approximating them with our calculators. 3
If a square root undoes an exponent of 2, what would undo an exponent of 3 a cube? A cube root. The cube root of a, =b only if b 3 =a What does 2 3 equal? As you can imagine, we can continue with this. There are fourth roots and fifth roots, etc. The nth root of a number a, is equal to a number b if b n = a. If n is an even number then we will need a to be a non negative number if is going to be a real number. But we don t need to be concerned if n is an odd number like 3 or 5. How to find on calculator: What if a is a variable rather than a number? to be completely precise. At this point, I am happy if you get to the x 3, but to be correct it really should be the absolute value of x 3. It is important when you get to calculus. 4
Graphing Square and Cube Root Functions If we are going to be graphing a function, we need real numbers. So with square root or any even root functions we have to be a little careful about the domain. we need x + 2 0. Solving for x gives us x 2 for the domain of the function. Table and graph: For cube root or any odd root functions, we don t need to worry about this because we can cube roots of negative numbers are still real numbers. So the domain of g x is all real numbers. Table and graph: 5
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