1. 1 Topic 6.1.4 Adding and Subtracting Polynomials Adding and Subtracting Polynomials

2. 2 Lesson 1.1.1 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. 10.0 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 use addition and subtraction of polynomials to solve more complicated problems. Adding and Subtracting Polynomials Topic 6.1.4 Key words: polynomial like terms

3. 3 Lesson 1.1.1 This Topic just contains one big example that uses both addition and subtraction of polynomials. Adding and Subtracting Polynomials Topic 6.1.4

4. 4 Example 1 Solution follows… Topic 6.1.4 Adding and Subtracting Polynomials Find a) the sum of and b) the difference between the perimeters of the rectangles shown below: 3 x 2 – 3 x – 2 2 x 2 – 3 x 2 + 2 x + 1 x 2 – 1 P1P1 P2P2 Solution The first thing you need to do is find and simplify expressions for each of the perimeters. Solution continues…

5. 5 Adding and Subtracting Polynomials Topic 6.1.4 Solution (continued) Perimeter of P 1 = (3 x 2 – 3 x – 2) + (3 x 2 – 3 x – 2) + (2 x 2 – 3) + (2 x 2 – 3) = 3 x 2 – 3 x – 2 + 3 x 2 – 3 x – 2 + 2 x 2 – 3 + 2 x 2 – 3 = 3 x 2 + 3 x 2 + 2 x 2 + 2 x 2 – 3 x – 3 x – 2 – 2 – 3 – 3 = 10 x 2 – 6 x – 10 Solution continues… Example 1 Find a) the sum of and b) the difference between the perimeters of the rectangles shown below: 3 x 2 – 3 x – 2 2 x 2 – 3 x 2 + 2 x + 1 x 2 – 1 P1P1 P2P2

6. 6 Perimeter of P 2 = ( x 2 + 2 x + 1) + ( x 2 + 2 x + 1) + ( x 2 – 1) + ( x 2 – 1) = x 2 + 2 x + 1 + x 2 + 2 x + 1 + x 2 – 1 + x 2 – 1 = x 2 + x 2 + x 2 + x 2 + 2 x + 2 x + 1 + 1 – 1 – 1 = 4 x 2 + 4 x Adding and Subtracting Polynomials Topic 6.1.4 Solution continues… Solution (continued) Example 1 Find a) the sum of and b) the difference between the perimeters of the rectangles shown below: 3 x 2 – 3 x – 2 2 x 2 – 3 x 2 + 2 x + 1 x 2 – 1 P2P2 P 1 = 10 x 2 – 6 x – 10

7. 7 a) Perimeter of P 1 + Perimeter of P 2 = (10 x 2 – 6 x – 10) + (4 x 2 + 4 x ) = 10 x 2 – 6 x – 10 + 4 x 2 + 4 x = 10 x 2 + 4 x 2 – 6 x + 4 x – 10 = 14 x 2 – 2 x – 10 Adding and Subtracting Polynomials Topic 6.1.4 Solution continues… Solution (continued) Example 1 Find a) the sum of and b) the difference between the perimeters of the rectangles shown below: 3 x 2 – 3 x – 2 2 x 2 – 3 x 2 + 2 x + 1 x 2 – 1 P 1 = 10 x 2 – 6 x – 10 P 2 = 4 x 2 + 4 x

8. 8 b) Perimeter of P 1 – Perimeter of P 2 = (10 x 2 – 6 x – 10) – (4 x 2 + 4 x ) = 10 x 2 – 6 x – 10 – 4 x 2 – 4 x = 10 x 2 – 4 x 2 – 6 x – 4 x – 10 = 6 x 2 – 10 x – 10 Adding and Subtracting Polynomials Topic 6.1.4 Solution (continued) Example 1 Find a) the sum of and b) the difference between the perimeters of the rectangles shown below: 3 x 2 – 3 x – 2 2 x 2 – 3 x 2 + 2 x + 1 x 2 – 1 P 1 = 10 x 2 – 6 x – 10 P 2 = 4 x 2 + 4 x

9. 9 Find the perimeter of each of these shapes. Lesson 1.1.1 Guided Practice Adding and Subtracting Polynomials Topic 6.1.4 Solution follows… 1. 2.3. 2(4(5 x 2 – 3 x + 1)) + 2(5( x 2 – x + 3)) = 50 x 2 – 34 x + 38 2(2 x + 4) + 2( x + 7) = 6 x + 22 6( x 2 + x ) + 3( x – 1) + 7( x 2 + x ) = 13 x 2 + 16 x – 3 4(5 x 2 – 3 x + 1) 5( x 2 – x + 3) 2( x + 7) 2 x +4 7( x 2 + x ) 3( x – 1)6( x 2 + x )

10. 10 Adding and Subtracting Polynomials Independent Practice Solution follows… Topic 6.1.4 Simplify each of these expressions. 1. 4 y – [3( y – x ) – 5( x + y )] 2. 3[2 y – 3( y – 1) + 2(– y – 1)] 3. 3( x – y ) – 2( x – 3 y ) 4. –2[( x – y ) – 2( x + 2 y )] – [(–2 x – y ) – ( x + 2 y )] 5. (–3 x 3 – x 2 y + 2 xy 2 – 2 y 3 ) – (–5 x 3 + x 2 y – 3 xy 2 – 3 y 3 ) + (– x 3 – 2 x 2 y + y 3 ) 6 y + 8 x –9 y + 3 x + 3 y 5 x + 13 y x 3 – 4 x 2 y + 5 xy 2 + 2 y 3

11. 11 Adding and Subtracting Polynomials Independent Practice Solution follows… Topic 6.1.4 6. Find the difference between the perimeters of the trapezoid and the triangle shown. x – 12 7. Simplify 5(4 x + 3) – 4(3 x – 1) + 3( x – 1). 11 x + 16 2 x 3 + x 2 – 6 x 2 x 3 – 2 x 2 + 4 x – 16 9. Simplify –2(3 x 3 – 2 x ) – 3(– x 3 + 7) – (2 x 2 – 5 x 3 – 5). 8. Simplify (8 x 3 – 5 x 2 – 2 x – 7) – (6 x 3 – 3 x 2 + x – 5) – (–3 x 2 + 3 x – 2). 3(2 x – 1) 3 x – 3 3( x – 2) 2(7 – x ) 5 x – 1 3(2 x – 1) 2(9 – x )

12. 12 Adding and Subtracting Polynomials Independent Practice Solution follows… Topic 6.1.4 10. Find the sum of the opposites of: –2 x 3 + 3 x 2 – 5 x + 1 and 3 x 3 – 2 x 2 + 3 x – 3 11. Find the difference between the opposites of: –2 x 2 – 3 x + 5 and 3 x 2 + 2 x – 4 12. Simplify –2(3 x 2 p – 2 x p + 1) + 4(2 x 2 p – x p – 3) – 5( x 2 p – 2 x p – 1). 5 x 2 + 5 x – 9 –3 x 2 p + 10 x p – 9 – x ³ – x 2 + 2 x + 2

13. 13 Adding and Subtracting Polynomials Independent Practice Solution follows… Topic 6.1.4 13. If the perimeter of the figure shown is 90 inches, what are the dimensions of the figure? 39 inches × 6 inches 3(2 x + 1) 2( x – 3)

14. 14 Topic 6.1.4 Round Up Adding and Subtracting Polynomials You’ve covered addition and subtraction of polynomials in just four short Topics. As you might expect, the next Section covers multiplication of polynomials, and then you’ll learn about division of polynomials in Section 6.3.