1 Topic Adding Polynomials

2 Lesson California Standards: 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. What it means for you: You’ll add polynomials and multiply a polynomial by a number. Adding Polynomials Topic Key words: polynomial like terms inverse

3 Lesson You saw in Topic that polynomials are just algebraic expressions with one or more terms. Adding Polynomials Topic Adding polynomials isn’t difficult at all. The only problem is that you can only add certain parts of each polynomial together.

4 Lesson The Opposite of a Polynomial Adding Polynomials Topic The opposite of a number is its additive inverse. The opposite of a positive number is its corresponding negative number, and vice versa. For example, –1 is the opposite of 1, and 1 is the opposite of –1. To find the opposite of a polynomial, you make the positive terms negative and the negative terms positive.

5 a) –2 x + 1 b) 5 x 2 – 3 x + 1 Adding Polynomials Example 1 Topic Find the opposites of the following polynomials: a) 2 x – 1 b) –5 x x – 1 Solution Solution follows…

6 Find the opposites of the following polynomials. Lesson Guided Practice Adding Polynomials Topic Solution follows… 1. 2 x –5 x – 1 3. x x – x 2 – 2 x x x – 86. –8 x 2 – 4 x x 4 – x 3 – 6 x x – x 4 – 6 x –2 x x 3 – 2 x –0.9 x 3 – 0.8 x 2 – 0.4 x – –1.4 x 3 – 0.8 x 2 – x 1 2 –2 x – 1 – x 2 – 5 x + 2 –3 x 2 – 4 x + 8 –4 x –5 x x 2 – x x x x + 1 –3 x x – 3 8 x x – 4 –8 x x 2 – 6 x x 4 – 3 x x x x 2 + x

7 Lesson Adding Polynomials Topic Adding polynomials consists of combining all like terms. There are a few ways of adding polynomials — one method is by collecting like terms and simplifying, another is through the vertical lining up of terms. The following Example explains these two methods.

8 (–5 x x – 1) + (6 x 2 – x + 3) + (5 x – 7) = –5 x x – x 2 – x x – 7 = –5 x x x – x + 5 x – – 7 = x x – 5 Adding Polynomials Example 2 Topic Find the sum of –5 x x – 1, 6 x 2 – x + 3, and 5 x – 7. Solution Method A — Collecting Like Terms and Simplifying Solution follows…

9 –5 x x – 1 Adding Polynomials Example 2 Topic Find the sum of –5 x x – 1, 6 x 2 – x + 3, and 5 x – 7. Solution Method B — Vertical Lining Up of Terms Solution continues… Both methods give the same solution. x x – x 2 – x x – 7

10 Lesson Multiplying a Polynomial by a Number Adding Polynomials Topic Multiplying a polynomial by a number is the same as adding the polynomial together several times.

11 Adding Polynomials Example 3 Topic Multiply x + 3 by 3. Solution Solution follows… ( x + 3) × 3 = ( x + 3) + ( x + 3) + ( x + 3) = x + x + x = 3 x + 9

12 Lesson Multiplying a Polynomial by a Number Adding Polynomials Topic The simple way to multiply a polynomial by a number is to multiply each term of the polynomial by the number. In other words, you multiply out the parentheses, using the distributive property of multiplication over addition.

13 Adding Polynomials Example 4 Topic Multiply x x – 4 by 3. Solution Solution follows… 3( x x – 4) = (3 × x 2 ) + (3 × 2 x ) – (3 × 4) = 3 x x – 12

14 Add these polynomials and simplify the resulting expressions. 13. (4 x 2 – 2 x – 1) + (3 x 2 + x – 10) 14. (11 x 4 – 5 x 3 – 2 x ) + (–7 x x x – 3) Lesson Guided Practice Adding Polynomials Topic Solution follows… x x – 3 –4 x 2 – 3 x + 5 –2 x 2 + x – 7 (4 x x 2 ) + (–2 x + x ) + (–1 – 10) = 7 x 2 – x – 11 c 3 + c c + 2– x 2 + x – –5 c 3 – 3 c c c 2 – c – 3 6 c 3 + c + 4 (11 x 4 – 7 x 4 ) + (–5 x x 3 ) + (–2 x + 5 x ) + (0 – 3) = 4 x 4 – 2 x x – 3

15 Lesson Guided Practice Adding Polynomials Topic Solution follows… Multiply these polynomials by y 2 – 7 y ( x 2 – 3 x + 3) 19. (– x 2 + x – 4)20. (2 x x + 2) 4(10 y 2 ) – 4(7 y ) + 4(5) = 40 y 2 – 28 y ( x 2 ) – 4(3 x ) + 4(3) = 4 x 2 – 12 x (– x 2 ) + 4( x ) – 4(4) = –4 x x – 16 4(2 x 2 ) + 4(5 x ) + 4(2) = 8 x x + 8

16 In Exercises 1-5, simplify the expression and state the degree of the resulting polynomial. Adding Polynomials Independent Practice Solution follows… Topic (2 x x – 7) + (7 x 2 – 3 x + 4) 2. ( x 3 + x – 4) + ( x 3 – 8) + (4 x 3 – 3 x – 1) 3. (– x 6 + x – 5) + (2 x 6 – 4 x – 6) + (–2 x x – 4) 4. (3 x 2 – 2 x + 7) + (4 x x – 8) + (–5 x x – 5) 5. (0.4 x 3 – 1.1) + (0.3 x 3 + x – 1.0) + (1.1 x x – 2.0) 9 x 2 – 3, degree 2 6 x 3 – 2 x – 13, degree 3 – x 6 – x – 15, degree 6 2 x x – 6, degree x x – 4.1, degree 3

17 In Exercises 6-7, simplify the expression and state the degree of the resulting polynomial. Independent Practice Solution follows… Topic – 4 a 3 – 2 a a 4 – 2 a 3 – 4 a a 4 – 4 a – c c – 0.48 –4.9 c 2 – 3.6 c c a 4 – 6 a 3 – 10 a + 4, degree 43.5 c 2 – 2.2 c , degree 2 Adding Polynomials

18 Independent Practice Solution follows… Topic Multiply each polynomial below by – a a – 2 9. – c c –6 x 3 – 4 x 2 + x – x x 2 – 4 x + 32 –16 a 2 – 12 a c 2 – 12 c – 4 24 x x 2 –4 x + 32 –96 x 3 – 64 x x – 128 Adding Polynomials

19 Independent Practice Solution follows… Topic Multiply each polynomial below by 2 a a 2 + a – –7 a a 2 – 5 a a a 2 – 16 a –14 a a 3 – 10 a a Adding Polynomials

20 Topic Round Up Adding polynomials can look hard because there can be several terms in each polynomial. The important thing is to combine each set of like terms, step by step. Adding Polynomials