Module 17.1

UnderstandingPolynomial Expressions

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What are polynomial expressions, and how do you simplify them?

A monomial is an expression consisting of a number, variable, orproduct of numbers and variables that have whole number exponents.

A monomial cannot have:• More than one term• A variable in its denominator• Fractional exponents

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The degree of a monomial is the sum of the exponents of the variables.A constant has a degree of 0.

Monomial Degree

8 0

𝑥 1

𝑥𝑦 2

6𝑥𝑦 2

2𝑥2 2

4𝑥𝑦2 3

9𝑎2𝑏2 4

21𝑎3𝑏2 5

152𝑝3𝑞3 6

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A polynomial can have any number of terms.

A binomial has 2 terms. For example: 𝒙 + 𝟕 𝑜𝑟 𝒙𝟐 + 𝟐𝒙

A trinomial has 3 terms. For example: 𝟏𝟔𝒓𝟐 + 𝟐𝟎𝒓 − 𝟕

Beyond that, it is a “Polynomial of n terms”, where n is that number. For example: 𝟐𝟐𝒓𝟒 + 𝟔𝒓𝟑 + 𝟐𝒓𝟐 − 𝟖𝒓 − 𝟒 is a Polynomial of 5 terms.

Polynomials are classified by both the number of terms they contain, and their degree.

Polynomials also have Degrees. The Degree of a polynomial is the largest of the degrees of the individual monomials.

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6th Degree Binomial

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P. 807The Standard Form Of A Polynomial Containing Only One Variable

The 1st term will have the greatest degree, the 2nd term will have the next greatest degree, and so on, until the final term, which will have the lowest degree.

When written in this form, the coefficient of the first term is called the Leading Coefficient.

A Quartic polynomialFor example:

Quartic Term

Linear Term

Quadratic Term

Constant Term

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Is the leading coefficient always the largest of the coefficients?

Can the leading coefficient be negative?

How Do You Simplify Polynomials?

3) Combine them by adding their coefficients

1) Rearrange it in descending degrees

Given

2) Identify like terms and group them together

Remember the invisible 1

Again - How Do You Simplify Polynomials?

1) Rearrange it in descending degrees.

2) Identify like terms and group them together.

Given

3) Combine them by adding their coefficients.

4) Include the terms that are left over, if any.

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2) Identify like terms and group them together

3) Combine them by adding their coefficients.

Terms can have more than one variable to be considered alike –as long as the powers are the same.

1) Rearrange in descending degrees

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Which properties of addition allow you to rearrange and add the coefficients of the expression?

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Given a polynomial expression describing a real-world situation and a specific value for the variable(s), evaluate the polynomial by substituting for the variable(s). Then interpret the result.

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