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A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. A monomial may be a constant or a single variable. The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0. A. 4 p 4 q 3.
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A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. A monomial may be a constant or a single variable. The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.
A. 4p4q3 Additional Example 1: Finding the Degree of a Monomial Find the degree of each monomial. Add the exponents of the variables: 4 + 3 = 7. The degree is 7. B. 7ed A variable written without an exponent has an exponent of 1. 1+ 1 = 2. The degree is 2. C. 3 There is no variable, but you can write 3 as 3x0. The degree is 0.
Remember! The terms of an expression are the parts being added or subtracted. See Lesson 1-7.
a. b. c. 1.5k2m 4x 2c3 Check It Out! Example 1 Find the degree of each monomial. Add the exponents of the variables: 2 + 1 = 3. The degree is 3. Add the exponents of the variables: 1 = 1. The degree is 1. Add the exponents of the variables: 3 = 3. The degree is 3.
A polynomialis a monomial or a sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree.
The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form. The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.
6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9 2 Degree 1 5 2 5 1 0 0 –7x5 + 4x2 + 6x + 9. The leading The standard form is coefficient is –7. Additional Example 2A: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. 6x – 7x5 + 4x2 + 9 Find the degree of each term. Then arrange them in descending order:
y2 + y6 – 3y y6 + y2 – 3y Degree 6 6 1 2 1 2 The standard form is y6 + y2 – 3y. The leading coefficient is 1. Additional Example 2B: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. y2 + y6 − 3y Find the degree of each term. Then arrange them in descending order:
Remember! A variable written without a coefficient has a coefficient of 1. y5 = 1y5
16 – 4x2 + x5 + 9x3 x5 + 9x3 – 4x2 + 16 Degree 0 2 5 3 5 3 2 0 The leading x5 + 9x3 – 4x2 + 16. The standard form is coefficient is 1. Check It Out! Example 2a Write the polynomial in standard form. Then give the leading coefficient. 16 – 4x2 + x5 + 9x3 Find the degree of each term. Then arrange them in descending order:
18y5 – 3y8 + 14y –3y8 + 18y5 + 14y Degree 8 1 5 8 5 1 The standard form is The leading –3y8 + 18y5 + 14y. coefficient is –3. Check It Out! Example 2b Write the polynomial in standard form. Then give the leading coefficient. 18y5 – 3y8 + 14y Find the degree of each term. Then arrange them in descending order:
Some polynomials have special names based on their degree and the number of terms they have.
Additional Example 3: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5n3 + 4n 5n3 + 4n is acubic binomial. Degree 3 Terms 2 B. –2x –2x is a linear monomial. Degree 1 Terms 1
6 is a constant monomial. Check It Out! Example 3 Classify each polynomial according to its degree and number of terms. a. x3 + x2 – x + 2 x3 + x2 – x + 2 is acubic polynomial. Degree 3 Terms 4 b. 6 Degree 0 Terms 1 –3y8 + 18y5+ 14yis an 8th-degree trinomial. c. –3y8 + 18y5+ 14y Degree 8 Terms 3
–144 + 200 76 Additional Example 4: Application A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? Substitute the time for t to find the lip balm’s height. –16t2 + 220 –16(3)2 + 200 The time is 3 seconds. –16(9) + 200 Evaluate the polynomial by using the order of operations.
Additional Example 5 Continued A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? After 3 seconds the lip balm will be 76 feet above the water.
–400 + 2006 1606 Check It Out! Example 4 What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 + 400t + 6. How high will this firework be when it explodes? Substitute the time for t to find the firework’s height. –16t2 + 400t + 6 –16(5)2 + 400(5) + 6 The time is 5 seconds. –16(25) + 400(5) + 6 –400 + 2000 + 6
Check It Out! Example 4 Continued What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes? When the firework explodes, it will be 1606 feet above the ground.
A rootof a polynomial in one variable is a value of the variable for which the polynomial is equal to 0.
3x2– 48 3x2– 48 3(4)2– 48 3(0)2– 48 3(16)– 48 3(0)– 48 48– 48 0– 48 0 –48 Additional Example 5: Identifying Roots of Polynomials Tell whether each number is a root of 3x2– 48. B. 0 A. 4 Substitute for x. Simplify. 4 is a root of 3x2– 48. 0 is not a root of 3x2– 48.
0 Additional Example 5: Identifying Roots of Polynomials Tell whether each number is a root of 3x2– 48. C. –4 3x2– 48 3(–4)2– 48 Substitute for x. 3(16)– 48 48– 48 Simplify. –4 is a root of 3x2– 48.
0 Check It Out! Example 5 Tell whether 1 is a root of 3x3 + x– 4. 3x3 + x– 4 3(1)3 + (1)– 4 Substitute for x. 3(1) + 1– 4 3 + 1– 4 Simplify. 1 is a root of 3x3 + x– 4.
