1

Topic 6.1.4Topic 6.1.4

Adding and Subtracting

Polynomials

Adding and Subtracting

Polynomials

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 PolynomialsAdding and Subtracting PolynomialsTopic

6.1.4

Key words:• polynomial• like terms

3

Lesson

1.1.1

This Topic just contains one big example that uses both addition and subtraction of polynomials.

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsTopic

6.1.4

4

Example 1

Solution follows…

Topic

6.1.4 Adding and Subtracting PolynomialsAdding and Subtracting Polynomials

Find a) the sum of and b) the difference between the perimeters of the rectangles shown below:

3x2 – 3x – 2

2x2 – 3

x2 + 2x + 1

x2 – 1 P1 P2

Solution

The first thing you need to do is find and simplify expressions for each of the perimeters.

Solution continues…

5

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsTopic

6.1.4

Solution (continued)

Perimeter of P1

= (3x2 – 3x – 2) + (3x2 – 3x – 2) + (2x2 – 3) + (2x2 – 3)

= 3x2 – 3x – 2 + 3x2 – 3x – 2 + 2x2 – 3 + 2x2 – 3

= 3x2 + 3x2 + 2x2 + 2x2 – 3x – 3x – 2 – 2 – 3 – 3

= 10x2 – 6x – 10 Solution continues…

Example 1

Find a) the sum of and b) the difference between the perimeters of the rectangles shown below:

3x2 – 3x – 2

2x2 – 3

x2 + 2x + 1

x2 – 1 P1 P2

6

Perimeter of P2

= (x2 + 2x + 1) + (x2 + 2x + 1) + (x2 – 1) + (x2 – 1)

= x2 + 2x + 1 + x2 + 2x + 1 + x2 – 1 + x2 – 1

= x2 + x2 + x2 + x2 + 2x + 2x + 1 + 1 – 1 – 1

= 4x2 + 4x

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsTopic

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:

3x2 – 3x – 2

2x2 – 3

x2 + 2x + 1

x2 – 1 P2P1 = 10x2 – 6x – 10

7

a) Perimeter of P1 + Perimeter of P2

= (10x2 – 6x – 10) + (4x2 + 4x)

= 10x2 – 6x – 10 + 4x2 + 4x

= 10x2 + 4x2 – 6x + 4x – 10

= 14x2 – 2x – 10

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsTopic

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:

3x2 – 3x – 2

2x2 – 3

x2 + 2x + 1

x2 – 1 P1 = 10x2 – 6x – 10 P2 = 4x2 + 4x

8

b) Perimeter of P1 – Perimeter of P2

= (10x2 – 6x – 10) – (4x2 + 4x)

= 10x2 – 6x – 10 – 4x2 – 4x

= 10x2 – 4x2 – 6x – 4x – 10

= 6x2 – 10x – 10

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsTopic

6.1.4

Solution (continued)

Example 1

Find a) the sum of and b) the difference between the perimeters of the rectangles shown below:

3x2 – 3x – 2

2x2 – 3

x2 + 2x + 1

x2 – 1 P1 = 10x2 – 6x – 10 P2 = 4x2 + 4x

9

Find the perimeter of each of these shapes.

Lesson

1.1.1

Guided Practice

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsTopic

6.1.4

Solution follows…

1.

2. 3.

2(4(5x2 – 3x + 1)) + 2(5(x2 – x + 3))= 50x2 – 34x + 38

2(2x + 4) + 2(x + 7)= 6x + 22

6(x2 + x) + 3(x – 1) + 7(x2 + x)= 13x2 + 16x – 3

4(5x2 – 3x + 1)

5(x2 – x + 3)

2(x + 7)

2x +4

7(x2 + x)

3(x – 1)6(x2 + x)

10

Adding and Subtracting PolynomialsAdding and Subtracting Polynomials

Independent Practice

Solution follows…

Topic

6.1.4

Simplify each of these expressions.

1. 4y – [3(y – x) – 5(x + y)]

2. 3[2y – 3(y – 1) + 2(– y – 1)]

3. 3(x – y) – 2(x – 3y)

4. –2[(x – y) – 2(x + 2y)] – [(–2x – y) – (x + 2y)]

5. (–3x3 – x2y + 2xy2 – 2y3) – (–5x3 + x2y – 3xy2 – 3y3) + (–x3 – 2x2y + y3)

6y + 8x

–9y + 3

x + 3y

5x + 13y

x3 – 4x2y + 5xy2 + 2y3

11

Adding and Subtracting PolynomialsAdding 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(4x + 3) – 4(3x – 1) + 3(x – 1).11x + 16

2x3 + x2 – 6x

2x3 – 2x2 + 4x – 169. Simplify –2(3x3 – 2x) – 3(–x3 + 7) – (2x2 – 5x3 – 5).

8. Simplify (8x3 – 5x2 – 2x – 7) – (6x3 – 3x2 + x – 5) – (–3x2 + 3x – 2).

3(2x – 1)

3x – 3

3(x

– 2)

2(7 – x)

5x – 1

3(2x

– 1

) 2(9 – x)

12

Adding and Subtracting PolynomialsAdding and Subtracting Polynomials

Independent Practice

Solution follows…

Topic

6.1.4

10. Find the sum of the opposites of: –2x3 + 3x2 – 5x + 1 and 3x3 – 2x2 + 3x – 3

11. Find the difference between the opposites of: –2x2 – 3x + 5 and 3x2 + 2x – 4

12. Simplify –2(3x2p – 2xp + 1) + 4(2x2p – xp – 3) – 5(x2p – 2xp – 1).

5x2 + 5x – 9

–3x2p + 10xp – 9

–x³ – x2 + 2x + 2

13

Adding and Subtracting PolynomialsAdding 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(2x + 1)

2(x – 3

)

14

Topic

6.1.4

Round UpRound Up

Adding and Subtracting PolynomialsAdding 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.