Foundations
Site: | Clare-Gladwin RESD |
Course: | Michigan Algebra II KHauck |
Book: | Foundations |
Printed by: | Guest user |
Date: | Saturday, November 23, 2024, 12:41 PM |
Description
Introduction
The Algebra I course covered linear, quadratic, exponential & polynomial functions. In this section we will further explore polynomial functions. A polynomial is composed of multiple terms that contain positive integer exponents.5x2 - 2x is a polynomial.
3x2 - 2x-2 is not a polynomial because it has a negative exponent.
$$ \sqrt{x-1} $$ is not a polynomial because it has a variable under the square root.
$$ \frac{5}{x^{3}+1^{}^{}} $$ is not a polynomial because it has an exponent in the denominator of a fraction.
$$4x^{0.3}$$ is not a polynomial because it has a decimal exponent.
Vocabulary
Terms of a polynomial are separated by addition. The example below is a polynomial with four terms.A variable is the unknown quantity represented by a letter, such as x in the example. A coefficient is the number in front of the variable in each term. If there is a subtraction sign, then the coefficient is negative. A constant is a number without a variable.
In this case, the coefficient of 4x3 is 4, the coefficient of 2x2 is 2, the coefficient of -3x is -3, and the constant is 1.
Degrees
Each term in the polynomial has a degree. In single variable polynomials the degree is the value of the exponent. In the example:4x3 has a degree of 3 and is called a cubic term or 3rd order term,
2x2 has a degree of 2 and is called the quadratic term or 2nd order term,
-3x has a degree of 1 and is called the linear term or 1st order term,
1 has a degree of 0 and is called the constant.
The degree of a polynomial is the same as the degree of the term with the largest exponent. Since the largest exponent in our example is 3, the polynomial is of degree 3, also called a "cubic" polynomial.
Multiple Variables
Polynomials and terms can have more than one variable. Here is another example of a polynomial:The positive integer exponents confirm this example is a polynomial. The polynomial has five terms. Let's look at each term more closely. When a term has multiple variables, the degree of the term is the sum of the exponents within the term.
t4 has a degree of 4, so it's a 4th order term,
-6s3t2 has a degree of 5 (3+2), so it's a 5th order term,
-12st has a degree of 2 (1+1), so it's a 2nd order term,
4s4 has a degree of 4, so it's a 4th order term,
-5 is a constant, so its degree is 0.
Since the largest degree of a term in this polynomial is 5, then this is a polynomial of degree 5 or a 5th order polynomial.
Monomials
A polynomial that has only one term is called a monomial (mono means one). A monomial can be a constant, a variable, or a product of a constant and one or more variables. You can see that each term in a polynomial is a monomial. A polynomial is the sum of monomials. Here are some examples of monomials:5
-6x4y2z
2t
Standard Form
Each term in a polynomial has a degree. The term with the largest degree determines the degree of the polynomial. In order for a polynomial to be in standard form, the term with the largest degree must be listed first followed by the remaining terms in descending order of degree.The following polynomials are in standard form:
4x4 - 3x3 + 2x2 - x +1
a4b3 - 2a3b3 + 3a4b - 5ab2 +2
The first term of a polynomial in standard form is called the leading term and the coefficient of the leading term is called the leading coefficient.
The first polynomial above has a leading term of 4x4 and a leading coefficient of 4.
The second polynomial above has a leading term of a4b3 and a leading coefficient of 1.
Combining Like Terms
A polynomial is in simplest form if all like terms have been combined. Like terms have the same variable(s) with the same exponents, but can have different coefficients.2x2y and 5x2y are like termsIf a polynomial has like terms, we simplify it by combining (adding) them.
6x2y and 6xy2 are not like terms
This sample polynomial is simplified by combining the like terms of 6xy and - 4xy giving us 2xy. We write the simplified polynomial as x2 + 2xy + y2.
Video Lesson
To learn how to combine like terms select the following link to Holt, Rinehart and Winston :
Combining Like Terms
Practice
Polynomial Foundation Worksheet/font>
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Answer Key
Polynomial Foundations Answer Key/font>
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Sources
Sources used in this book:Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project
CK12 Foundation, "Factoring Polynomials: More on Probability." 11/6/2009.http://ck12.org/flexr/viewer/962d35c81e3e9e1f4a10fdb462386484/(accessed 12/3/2009).
Holt Rinehart & Winston, "Exponents and Polynomials Homework Help Online." http://my.hrw.com/math06_07/nsmedia/homework_help/alg1/alg1_ch07_06_homeworkhelp.html (accessed 6/4/2010).