1. Introduction Expressions can be used to represent quantities when those quantities are a sum of other values. When there are unknown values in the sum, variables are used. Expressions with variables raised to powers are often written in a standard way. This allows expressions to be more easily compared. 1 2.1.1: Structures of Expressions

2. Key Concepts A factor is one of two or more numbers or expressions that when multiplied produce a given product. For example, in the expression 2a, 2 and a are each a factor. An exponential expression is an expression that contains a base raised to a power or exponent. For example, a 2 is an exponential expression: a is the base and 2 is the power. 2 2.1.1: Structures of Expressions

3. Key Concepts, continued A polynomial is an expression that contains variables and/or numeric quantities where the variables are raised to integer powers greater than or equal to 0. For example, b 6 + 2a – 4 is a polynomial. The terms in the polynomial b 6 + 2a – 4 are b 6, 2a, and –4. The degree of a polynomial is the greatest exponent of the variable. 3 2.1.1: Structures of Expressions

4. Key Concepts, continued A polynomial function is a function written in the following form: where a 1 is a rational number, a n ≠ 0, and n is a nonnegative integer and the highest degree of the polynomial. The function f(x) = 8x 5 + 7x 3 + x 2 + x is a polynomial function. When a term is the product of a number and a variable, the numeric portion is called the coefficient of the term. 4 2.1.1: Structures of Expressions

5. Key Concepts, continued When the term is just a variable, the coefficient is always 1. The terms in a polynomial are ordered by the power of the variables, with the largest power listed first. This is known as descending order. A term with no variable, called a constant term, is listed after any terms that have a variable. Constant terms can be rewritten as the product of a numeric value and a variable raised to the power of 0. For example, in the polynomial expression 4x 5 + 9x 4 + 3x + 12, 12 could be thought of as 12x 0. 5 2.1.1: Structures of Expressions

6. Key Concepts, continued The leading coefficient of a polynomial is the coefficient of the term with the highest power. For example, in the polynomial expression 4x 5 + 9x 4 + 3x + 12, the term with the highest power is 4x 5 ; therefore, the leading coefficient is 4. 6 2.1.1: Structures of Expressions

7. Common Errors/Misconceptions incorrectly ordering terms with different powers combining terms in an expression that have variables raised to different powers 7 2.1.1: Structures of Expressions

8. Guided Practice Example 2 Identify the terms in the expression –2x 8 + 3x 2 – x + 11, and note the coefficient, variable, and power of each term. 8 2.1.1: Structures of Expressions

9. Guided Practice: Example 2, continued 1.Rewrite any subtraction using addition. Subtraction can be rewritten as adding a negative quantity. –2x 8 + 3x 2 – x + 11 = –2x 8 + 3x 2 + (–x) + 11 9 2.1.1: Structures of Expressions

10. Guided Practice: Example 2, continued 2.List the terms being added. There are four terms in the expression: –2x 8, 3x 2, –x, and 11. 10 2.1.1: Structures of Expressions

11. Guided Practice: Example 2, continued 3.Identify the coefficient, variable, and power of each term. The coefficient is the number being multiplied by the variable. The variable is the quantity represented by a letter. The power is the value of the exponent of the variable. The term –2x 8 has a coefficient of –2, a variable of x, and a power of 8. 11 2.1.1: Structures of Expressions

12. Guided Practice: Example 2, continued The term 3x 2 has a coefficient of 3, a variable of x, and a power of 2. The term –x has a coefficient of –1, a variable of x, and a power of 1. The term 11 is a constant; it contains only a numeric value. 12 2.1.1: Structures of Expressions ✔

13. Guided Practice: Example 2, continued 13 2.1.1: Structures of Expressions

14. Guided Practice Example 3 Write a polynomial function in descending order that contains the terms –x, 10x 5, 4x 3, and x 2. Determine the degree of the polynomial function. 14 2.1.1: Structures of Expressions

15. Guided Practice: Example 3, continued 1.Identify the power of the variable of each term. The power of the variable is the value of the exponent of the variable. When no power is shown, the power is 1. Therefore, the power of the term –x is 1. The power of the term 10x 5 is 5. The power of the term 4x 3 is 3. The power of the term x 2 is 2. 15 2.1.1: Structures of Expressions

16. Guided Practice: Example 3, continued 2.Order the terms in descending order using the powers of the exponents. The term with the highest power is listed first, then the term with the next highest power, and so on, with a constant listed last. 10x 5, 4x 3, x 2, –x 16 2.1.1: Structures of Expressions

17. Guided Practice: Example 3, continued 3.Sum the terms to write the polynomial function of the given variable. The variable in each term is x, so the function will be a function of x, written f(x). Write f(x) as the sum of the terms in descending order, with the term that has the highest power listed first. f(x) = 10x 5 + 4x 3 + x 2 + (–x) f(x) = 10x 5 + 4x 3 + x 2 – x 17 2.1.1: Structures of Expressions

18. Guided Practice: Example 3, continued 4.Determine the degree of the polynomial. The degree of a polynomial is the highest power of the variable. The term with the highest power is listed first in the function, 10x 5, and x has a power of 5. The degree of the polynomial function f(x) is 5. 18 2.1.1: Structures of Expressions ✔

19. Guided Practice: Example 3, continued 19 2.1.1: Structures of Expressions