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 1 4 2 -1

Lesson 7.6 Polynomials

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.

Words to Know Monomial- Polynomial- Binomials - Two terms Trinomials Single term consisting of numbers, variables, and whole number exponents. single Polynomial- An expression consisting of two or more monomial terms that can’t be combined. many Types of Polynomials Binomials - Two terms No negative powers No variables in the denominator Trinomials - Three terms

Identify each expression as a monomial, binomial, trinomial, or polynomial. (More than one may apply.) Binomial , Polynomial Polynomial Trinomial , Polynomial Binomial , Polynomial Monomial Trinomial , Polynomial

Standard Form of the Polynomial - Powers of the variable written in decreasing order. Non-standard Form Standard Form

You try! Write the polynomials below in standard form.

The degree of a polynomial is the degree of the term with the greatest degree. When written in standard form, the coefficient of the first term is called the leading coefficient.

Write the polynomial 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 leadingcoefficient is –7.

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

Now you try… 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.

Simplifying Polynomials - Simplified by combining like terms. Simplify the polynomials below. Write them in standard form.

Simplify the following polynomials. Write in standard form.

Simplify the following polynomials. Write in standard form. Answers

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

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.

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 The time is 3 seconds. –16(3)2 + 200 –16(9) + 200 Evaluate the polynomial by using the order of operations. –144 + 200 76 feet

Find the degree of each polynomial. 1. 7a3b2 – 2a4 + 4b – 15 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 Tell whether each number is a root of 3p2 – 8 + 4. 5. 2 yes 6. –2 no

Lesson Quiz: Part III 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