Preview Warm Up California Standards Lesson Presentation

Warm Up Evaluate each expression for the given value of x. 1. 2x + 3; x = 2 2. x2 + 4; x = –3 3. –4x – 2; x = –1 4. 7x2 + 2x; x = 3 Identify the coefficient in each term. 5. 4x3 6. y3 7. 2n7 8. –s4 7 13 2 69 4 1 2 –1

California Standards Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques.

Vocabulary monomial degree of a monomial polynomial degree of a polynomial standard form of a leading coefficient quadratic cubic binomial trinomial roots

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.

Additional Example 1: Finding the Degree of a Monomial Find the degree of each monomial. A. 4p4q3 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.

The terms of an expression are the parts being added or subtracted The terms of an expression are the parts being added or subtracted. See Lesson 1-7. Remember!

Check It Out! Example 1 Find the degree of each monomial. a. 1.5k2m Add the exponents of the variables: 2 + 1 = 3. The degree is 3. b. 4x Add the exponents of the variables: 1 = 1. The degree is 1. c. 2c3 Add the exponents of the variables: 3 = 3. The degree is 3.

A polynomial is a monomial or a sum or difference of monomials A polynomial is 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 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.

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: 6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9 Degree 1 5 2 –7x5 + 4x2 + 6x + 9. The standard form is The leading coefficient is –7.

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: y2 + y6 – 3y y6 + y2 – 3y Degree 2 6 1 The standard form is The leading coefficient is 1. y6 + y2 – 3y.

A variable written without a coefficient has a coefficient of 1. Remember! y5 = 1y5

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: 16 – 4x2 + x5 + 9x3 x5 + 9x3 – 4x2 + 16 Degree 2 5 3 The standard form is The leading coefficient is 1. x5 + 9x3 – 4x2 + 16.

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: 18y5 – 3y8 + 14y –3y8 + 18y5 + 14y Degree 5 8 1 The standard form is The leading coefficient is –3. –3y8 + 18y5 + 14y.

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 a cubic binomial. Degree 3 Terms 2 B. –2x –2x is a linear monomial. Degree 1 Terms 1

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 a cubic polynomial. Degree 3 Terms 4 b. 6 6 is a constant monomial. Degree 0 Terms 1 –3y8 + 18y5 + 14y is an 8th-degree trinomial. c. –3y8 + 18y5 + 14y Degree 8 Terms 3

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. –144 + 200 76

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.

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 –400 + 2006 1606

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 root of a polynomial in one variable is a value of the variable for which the polynomial is equal to 0.

Additional Example 5: Identifying Roots of Polynomials Tell whether each number is a root of 3x2 – 48. A. 4 B. 0 3x2 – 48 3(4)2 – 48 3x2 – 48 3(0)2 – 48 Substitute for x. 3(16) – 48 48 – 48 3(0) – 48 0 – 48 Simplify.  –48  4 is a root of 3x2 – 48. 0 is not a root of 3x2 – 48.

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.

 Check It Out! Example 5 Tell whether 1 is a root of 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.

Lesson Quiz: Part I Find the degree of each polynomial. 1. 7a3b2 – 2a4 + 4b – 15 2. 25x2 – 3x4 Write each polynomial in standard form. Then give the leading coefficient. 3. 24g3 + 10 + 7g5 – g2 4. 14 – x4 + 3x2 5 4 7g5 + 24g3 – g2 + 10; 7 –x4 + 3x2 + 14; –1

Lesson Quiz: Part II Classify each polynomial according to its degree and number of terms. 5. 18x2 – 12x + 5 quadratic trinomial 6. 2x4 – 1 quartic binomial 7. The polynomial 3.675v + 0.096v2 is used to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat, dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727.65 ft

Lesson Quiz: Part IIl Tell whether each number is a root of 3p2 – 8 + 4. 8. 2 yes 9. –2 no