Polynomials
Learning Objectives
- Identify monomials, binomials, and polynomials.
- Write polynomials to describe real world situations.
Introduction
Algebraic expressions containing one or more termsA value in a sequence--the first value in a sequence is the `1"st"` term, the second value is the `2"nd"` term, and so on; a term is also any of the monomials that make up a polynomial. are called polynomialsA monomial or sum of monomials, like `4x^2 + 3x - 10`. . There are several kinds of polynomials, based on how many terms they have. For example, monomialsA number, a variable, or a product of a number and one or more variables with whole number exponents, such as `-5`, `x`, and `8xy^3`. are polynomials with one term (“mono-” is a prefix meaning one).
Polynomials are useful because they can be written to represent real world situations and to find solutions to actual problems.
Binomials and Polynomials
The prefix “poly-” means many. A polynomial is a collection of one or more monomials—it may have many terms. Polynomials that have two terms are called binomialsA sum of two monomials, such as `3x^2 + 7`. . The prefix “bi-” means two—you've probably noticed that a bicycle is a cycle with two wheels.
There are rules for writing polynomials. A polynomial cannot have a variable in the denominator or a negative exponent, since monomials must have only whole number exponents. Polynomials are generally written so that the powers of one variable are in descending order. For example, `3x^2 + 5 + 2x^3 + 8x` would be written `2x^3 + 3x^2 + 8x + 5`. The table below illustrates some examples of monomials, binomials, and polynomials.
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Monomials
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Binomials
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Other Polynomials
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`15`
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`3y + 13`
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`x + y + z`
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`1/2x`
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`4p - 7r`
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`3x^2 + 2x - 9`
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`-4y^3`
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`3x^2+5/8y^2`
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`x^2 + 3xy - 2y^2`
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`16m^2n^4`
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`3xy + 14x^2y^3`
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`5y^4 + 3p^3 - 6r^2 + 2x`
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Which of the following expressions are polynomials?
`2x^2 - 3y^3`
`14`
`(x+y)/z`
A) Only `2x^2 - 3y^3` and `(x+y)/z` are polynomials.
B) Only `2x^2 - 3y^3` and `14` are polynomials.
C) Only `2x^2 - 3y^3` is a polynomial.
D) None of the expressions is a polynomial.
A) `(x+y)/z` is not a polynomial because it has a variable in the denominator. The correct answer is `2x^2 - 3y^3` and `14`.
B) Correct. `(x+y)/z` is not a polynomial because it has a variable in the denominator.
C) Incorrect. `14` is also a monomial. A monomial is a type of polynomial because a polynomial can contain one or more terms, so the monomial having one term then also qualifies as a polynomial. The correct answer is `2x^2 - 3y^3` and `14`.
D) Incorrect. `2x^2 - 3y^3` and `14` are polynomials. `(x+y)/z` is not a polynomial because it has a variable in the denominator.
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Using Polynomials to Model Real World Situations
Writing a polynomial to represent a situation can help us answer questions and find solutions. Consider the following:
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Example
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Problem
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Sarina has published a book and wishes to send it to `200` readers. She has prepared `200` packages to mail. The cost of labor and materials to prepare for shipment is `$95.00`. The rates for shipping are:
`$16.50` per package international
`$4.90` per package domestic (in the U.S.)
Write a simplified polynomial to express the total cost to distribute her book if she ships `p` books within the U.S. and the rest internationally.
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`p = ` domestic packages
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Think about what is known and unknown in the problem. Use the variable, `p`, to represent the number of packages that Sarina will ship to domestic addresses.
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`200 - p =` international packages
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Since the total number of packages is `200`, and `p` are shipped domestically, the remaining books to be shipped outside the U.S. can be represented as `200-p`.
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packaging ` = 95`
U.S. shipping ` = 4.90p`
Int'l shipping `= 16.50(200 - p)`
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There are `3` elements to the cost of distributing the books.
The expression to represent shipping cost to each area comes from multiplying cost per book by the number of books.
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`95 + 4.90p + 16.50(200 - p)`
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Add the three elements of the cost together to write an expression representing the total cost to distribute the books.
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`95 + 4.90p+3300 - 16.50p`
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Use the distributive property.
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`3395 - 11.60p`
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Combine like terms.
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Answer
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`3395 - 11.60p`
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Sarina can use this polynomial to find out the cost for shipping `200` books when `p` of them are shipped within the U.S. (domestically).
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Example
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Problem
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A carpet designer creates a carpet that uses four colors according to the pattern and dimensions below. Express the area of the carpet as a polynomial.
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Red:
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`x*x=x^2`
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Find the area of each colored piece of carpet by multiplying the length by the width.
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Blue:
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`2y(x+3y)=2xy+6y^2`
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White:
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`2y*x=2xy`
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Green:
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`x(x+3y)=x^2+3xy`
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`x^2+2xy+6y^2+2xy+x^2+3xy`
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Find the area of the whole carpet by combining the areas of the four pieces.
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`2x^2+7xy+6y^2`
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Combine like terms.
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Answer
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`2x^2+7xy+6y^2`
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Summary
One of the powers of algebra is in representing aspects of the world with algebraic expressions in order to learn more about them. An expression that combines one or more terms to describe a situation is called a polynomial. Binomials, which are polynomials with two terms, and monomials, which are polynomials with one term, are two types of polynomials. By definition, polynomials do not have variables in the denominator or negative exponents.