Splash Screen Unit 8 Quadratic Expressions and Equations EQ: How do you use addition, subtraction, multiplication, and factoring of polynomials in order to simplify rational expressions?

Splash Screen Essential Question: How do you identify, add, and subtract polynomials? Lesson 1 Adding and Subtracting Polynomials

Then/Now You identified monomials and their characteristics. Write polynomials in standard form. Add and subtract polynomials. EQ: How do you identify, add, and subtract polynomials?

Vocab polynomial binomial trinomial degree of a monomial degree of a polynomial standard form of a polynomial leading coefficient

Vocab REMEMBER THIS DEFINITION? monomial - a polynomial with only one term; it contains no addition or subtraction. It can be a number, a variable, or a product of numbers and/or more variables

Vocab polynomial - an algebraic expression that is a monomial or the sum or difference of two or more monomials binomial - a polynomial with two unlike terms trinomial - a polynomial with three unlike terms (e.g., 7a + 4b + 9c). Each term is a monomial, and the monomials are joined by an addition symbol (+) or a subtraction symbol (–). It is considered an algebraic expression.

Vocab degree of a monomial - the sum of all the exponents of the variables, including the implicit exponents of 1 for the variables which appear without exponent. example:__________________________ degree of a polynomial - the value of the greatest exponent in a polynomial.

Vocab standard form of a polynomial – writing the terms of a polynomial in descending order (greatest to least degree). leading coefficient - the coefficient of the first term of a polynomial when written in descending order.

Over Chapter 7 5-Minute Check 1 A.yes; monomial B.yes; binomial C.yes; trinomial D.not a polynomial Determine whether – 8 is a polynomial. If so, identify it as a monomial, binomial, or trinomial.

Over Chapter 7 5-Minute Check 2 A.yes; monomial B.yes; binomial C.yes; trinomial D.not a polynomial

Over Chapter 7 5-Minute Check 4 A.6 B.5 C.4 D.3 What is the degree of the polynomial 5ab 3 + 4a 2 b + 3b 5 – 2?

Example 1 Identify Polynomials State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial A. State whether 3x 2 + 2y + z is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial B. State whether 4a 2 – b –2 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial C. State whether 8r – 5s is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial D. State whether 3y 5 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

Example 2 Standard Form of a Polynomial A. Write 9x 2 + 3x 6 – 4x in standard form. Identify the leading coefficient. Answer: 3x 6 + 9x 2 – 4x; the leading coefficient is 3. Step 2Write the terms in descending order. Step 1Find the degree of each term. Degree: 261 Polynomial:9x 2 + 3x 6 – 4x

Example 2 Standard Form of a Polynomial B. Write y + 6xy + 8xy 2 in standard form. Identify the leading coefficient. Answer: 8xy 2 + 6xy + 5y + 12 ; the leading coefficient is 8. Step 2Write the terms in descending order. Step 1Find the degree of each term. Degree: 0123 Polynomial: y + 6xy + 8xy 2

Example 2 A.3x 7 + 9x 4 – 4x 2 – 34x B. 9x 4 + 3x 7 – 4x 2 – 34x C. –4x 2 + 9x 4 + 3x 7 – 34x D.3x 7 – 4x 2 + 9x 4 – 34x A. Write – 34x + 9x 4 + 3x 7 – 4x 2 in standard form.

Example 2 A.–72 B. 8 C. –6 D.72 B. Identify the leading coefficient of 5m + 21 –6mn + 8mn 3 – 72n 3 when it is written in standard form.

Example 3 Add Polynomials A. Find (7y 2 + 2y – 3) + (2 – 4y + 5y 2 ). Horizontal Method = 7y 2 + 2y – – 4y + 5y 2 Write without parentheses = (7y 2 + 5y 2 ) + [2y + (–4y) + [(–3) + 2]Group like terms. = 12y 2 – 2y – 1Combine like terms. Answer: 12y 2 – 2y – 1

Example 3 Add Polynomials Vertical Method Answer: 12y 2 – 2y – 1 7y 2 + 2y – 3 (+) 5y 2 – 4y + 2 Notice that terms are in descending order with like terms aligned. 12y 2 – 2y – 1 A. Find (7y 2 + 2y – 3) + (2 – 4y + 5y 2 ).

Example 3 Add Polynomials B. Find (4x 2 – 2x + 7) + (3x – 7x 2 – 9). = 4x 2 – 2x x – 7x 2 – 9Write without parentheses = [4x 2 + (–7x 2 )] + [(–2x) + 3x] + [7 + (–9)]Group like terms. = – 3x 2 + x – 2Combine like terms. Answer: – 3x 2 + x – 2 Horizontal Method

Example 3 Add Polynomials Answer: – 3x 2 + x – 2 Align and combine like terms. 4x 2 – 2x + 7 (+) –7x 2 + 3x – 9 –3x 2 + x – 2 Vertical Method B. Find (4x 2 – 2x + 7) + (3x – 7x 2 – 9).

Example 3 A.– 2x 2 + 5x + 3 B.8x 2 + 6x – 4 C.2x 2 + 5x + 4 D.– 15x 2 + 6x – 4 A. Find (3x 2 + 2x – 1) + (– 5x 2 + 3x + 4).

Example 3 A.5x 2 + 3x – 6 B.4x 3 + 5x 2 + 3x – 6 C.7x 3 + 5x 2 + 3x – 6 D. 7x 3 + 6x 2 + 3x – 6 B. Find (4x 3 + 2x 2 – x + 2) + (3x 2 + 4x – 8).

Example 4 Subtract Polynomials A. Find (6y 2 + 8y 4 – 5y) – (9y 4 – 7y + 2y 2 ). Subtract 9y 4 – 7y + 2y 2 by adding its additive inverse. (6y 2 + 8y 4 – 5y) – (9y 4 – 7y + 2y 2 ) = (6y 2 + 8y 4 – 5y) + (– 9y 4 + 7y – 2y 2 ) = 6y 2 + 8y 4 – 5y + (– 9y 4 ) + 7y + (– 2y 2 ) = [8y 4 + (– 9y 4 )] + [6y 2 + (– 2y 2 )] + (– 5y + 7y) = –y 4 + 4y 2 + 2y Answer: – y 4 + 4y 2 + 2y Horizontal Method

Example 4 Subtract Polynomials Align like terms in columns and subtract by adding the additive inverse. Answer: – y 4 + 4y 2 + 2y 8y 4 + 6y 2 – 5y (–)9y 4 + 2y 2 – 7y Add the opposite. 8y 4 + 6y 2 – 5y (+) –9y 4 – 2y 2 + 7y –y 4 + 4y 2 + 2y Vertical Method A. Find (6y 2 + 8y 4 – 5y) – (9y 4 – 7y + 2y 2 ).

Example 4 Subtract Polynomials B. Find (6n n 3 + 2n) – (4n – 3 + 5n 2 ). Answer: 11n 3 + n 2 – 2n + 3 Subtract 4n 4 – 3 + 5n 2 by adding the additive inverse. (6n n 3 + 2n) – (4n – 3 + 5n 2 ) = (6n n 3 + 2n) + (– 4n + 3 – 5n 2 ) = 6n n 3 + 2n + (– 4n) (– 5n 2 ) = 11n 3 + [6n 2 + (– 5n 2 )] + [2n + (– 4n)] + 3 = 11n 3 + n 2 – 2n + 3 Horizontal Method

Example 4 Subtract Polynomials Align like terms in columns and subtract by adding the additive inverse. Answer: 11n 3 + n 2 – 2n n 3 + 6n 2 + 2n + 0 (–) 0n 3 + 5n 2 + 4n – 3 Add the opposite. 11n 3 + 6n 2 + 2n + 0 (+) 0n 3 – 5n 2 – 4n n 3 + n 2 – 2n + 3 Vertical Method B. Find (6n n 3 + 2n) – (4n – 3 + 5n 2 ).

Example 4 A.2x 2 + 7x 3 – 3x 4 B.x 4 – 2x 3 + x 2 C.x 2 + 8x 3 – 3x 4 D.3x 4 + 2x 3 + x 2 A. Find (3x 3 + 2x 2 – x 4 ) – (x 2 + 5x 3 – 2x 4 ).

Example 4 A.2y 4 – 2y 2 – 11 B.2y 4 + 5y 3 + 3y 2 – 11 C.2y 4 – 5y 3 + 3y 2 – 11 D.2y 4 – 5y 3 + 3y B. Find (8y 4 + 3y 2 – 2) – (6y 4 + 5y 3 + 9).

End of the Lesson Assignment: Worksheet Essential Question: How do you identify, add, and subtract polynomials?

Example 5 Add and Subtract Polynomials A. VIDEO GAMES The total amount of toy sales T (in billions of dollars) consists of two groups: sales of video games V and sales of traditional toys R. In recent years, the sales of traditional toys and total sales could be modeled by the following equations, where n is the number of years since R = 0.46n 3 – 1.9n 2 + 3n + 19 T = 0.45n 3 – 1.85n n A. Write an equation that represents the sales of video games V. video games + traditional toys = total toy sales V + R = T V = T – R

Example 5 Add and Subtract Polynomials Find an equation that models the sales of video games V. V =T – R Subtract the polynomial for R from the polynomial for T. 0.45n 3 – 1.85n n+22.6 (–) 0.46n 3 – 1.9n 2 +3n+19 Answer: V = – 0.01n n n Add the opposite. 0.45n 3 –1.85n n (+) –0.46n n 2 –3n– 19 –0.01n n n+ 3.6

Example 5 Add and Subtract Polynomials B. Use the equation to predict the amount of video game sales in the year Answer: The amount of video game sales in 2009 will be billion dollars. The year 2009 is 2009 – 2000 or 9 years after the year Substitute 9 for n. V= – 0.01(9) (9) (9) = – = V = – 0.01n n n + 3.6

Example 5 A.50x 2 – 50x B.–50x 2 – 50x C.250x x D.50x x A. BUSINESS The profit a business makes is found by subtracting the cost to produce an item C from the amount earned in sales S. The cost to produce and the sales amount could be modeled by the following equations, where x is the number of items produced. C = 100x x – 300 S = 150x x Find an equation that models the profit.

Example 5 A.$500 B.$30 C.$254,000 D.$44,000 B. Use the equation 50x 2 – 50x to predict the profit if 30 items are produced and sold.

End of the Lesson Assignment: Worksheet #2 Essential Question: How do you identify, add, and subtract polynomials?