14-3 Adding Polynomials Warm Up Combine like terms. 1. 9x + 4x2. –3y + 7y 3. 7n + (–8n) + 12n...
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Transcript of 14-3 Adding Polynomials Warm Up Combine like terms. 1. 9x + 4x2. –3y + 7y 3. 7n + (–8n) + 12n...
14-3 Adding Polynomials
Warm UpCombine like terms.
1. 9x + 4x 2. –3y + 7y3. 7n + (–8n) + 12n
Find the perimeter of each rectangle.
4. a 10 ft by 12 ft rectangle5. a 5 m by 8 m rectangle
Simplify.
6. 3(2x2 – x) + x2+ 1
13x 4y11n
44 ft26 m
7x2 – 3x + 1
14-3 Adding Polynomials
Learn to add and subtract polynomials.
14-3 Adding Polynomials
Example 1A: Adding Polynomials Horizontally
Add.
(5x3 + x2 + 2) + (4x3 + 6x2)
(5x + x + 2) + (4x + 6x )3 2 3 2
5x + x + 2 + 4x + 6x 3 2 3 2
9x + 7x + 23 2
Associative Property
Combine like terms.
14-3 Adding Polynomials
Example 1B: Adding Polynomials Horizontally
Add.
(6x3+ 8y2 + 5xy) + (4xy – 2y2)
(6x3+ 8y2+ 5xy) + (4xy – 2y2)
6x3 + 8y2 + 5xy + 4xy – 2y2
6x + 6y + 9xy3 2
Associative Property
Combine like terms.
14-3 Adding Polynomials
Example 1C: Adding Polynomials Horizontally
Add.
(3x2y – 5x) + (4x + 7) + 6x2y
(3x2y – 5x) + (4x + 7) + 6x2y
9x2y – x + 7
Associative Property
Combine like terms.
3x2y – 5x + 4x + 7 + 6x2y
14-3 Adding Polynomials
Example 2A
Add.
(3y4 + y2 + 6) + (5y4 + 2y2)
(3y + y + 6) + (5y + 2y )4 2 4 2
3y + y + 6 + 5y + 2y 4 2 4 2
8y + 3y + 64 2
Associative Property
Combine like terms.
14-3 Adding Polynomials
Example 2B
Add.
(9x3 + 6p2 + 3xy) + (8xy – 3p2)
(9x + 6p2 + 3xy) + (8xy – 3p2)3
9x + 6p + 3xy + 8xy – 3p 3 2 2
9x3+ 3p2 + 11xy
Associative Property
Combine like terms.
14-3 Adding Polynomials
Example 2C
Add.
(3z2w – 5x) + (2x + 8) + 6z2w
(3z2w – 5x) + (2x + 8) + 6z2w
9z2w – 3x + 8
Associative Property
Combine like terms.
3z2w – 5x + 2x + 8 + 6z2w
14-3 Adding Polynomials
You can also add polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms. If the terms are rearranged, remember to keep the correct sign with each term.
14-3 Adding Polynomials
Example 3: Adding Polynomials Vertically
Add.
A. (4x2 + 2x + 11) + (2x2 + 6x + 9)
4x2 + 2x + 11
+ 2x2 + 6x + 9
6x2 + 8x + 20 Combine like terms.
Place like terms in columns.
14-3 Adding Polynomials
Example 3: Adding Polynomials Vertically
Add.
B. (3mn2 – 6m + 6n) + (5mn2 + 2m – n)
C. (–x2y2 + 5x2) + (–2y2 + 2) + (x2 + 8)
+ 5mn2 + 2m – n
8mn2 – 4m + 5n
–2y2 + 2
+ x2 + 8
–x2y2 + 6x2 – 2y2 + 10
3mn2 – 6m + 6n
–x2y2 + 5x2
Combine like terms.
Place like terms in columns.
Combine like terms.
Place like terms in columns.
14-3 Adding Polynomials
Example 4
Add.
A. (6x2 + 6x + 13) + (3x2+ 2x + 4)
6x2 + 6x + 13
+ 3x2 + 2x + 4
9x2 + 8x + 17 Combine like terms.
Place like terms in columns.
14-3 Adding Polynomials
Example 4
Add.
B. (4mn2 + 6m + 2n) + (2mn2 – 2m – 2n)
C. (x2y2 – 5x2) + (2y2 – 2) + (x2)
+ 2mn2 – 2m – 2n
6mn2 + 4m
4mn2 + 6m + 2n
Combine like terms.
Place like terms in columns.
2y2 – 2
+ x2
x2y2 – 4x2 + 2y2 – 2
x2y2 – 5x2
Combine like terms.
Place like terms in columns.
14-3 Adding Polynomials
Subtraction is the opposite of addition. To subtract a polynomial, you need to find itsopposite.
14-3 Adding Polynomials
Find the opposite of each polynomial.
Example 1: Finding the Opposite of a Polynomial
A. 8x3y4z2
–(8x3y4z2)
–8x3y4z2 Distributive Property.
B. –3x4 + 8x2
–(–3x4 + 8x2)
3x4 – 8x2 Distributive Property.
14-3 Adding Polynomials
Find the opposite of the polynomial.
Additional Example 1: Finding the Opposite of a Polynomial
C. 9a6b4 + a4b2 – 1
–(9a6b4 + a4b2 – 1)
–9a6b4 – a4b2 + 1 Distributive Property.
14-3 Adding Polynomials
To subtract a polynomial, add its opposite.
14-3 Adding Polynomials
Subtract.
Example 1: Subtracting Polynomials Horizontally
A. (5x2 + 2x – 3) – (3x2 + 8x – 4)
= (5x2 + 2x – 3) + (–3x2 – 8x + 4)
= 5x2 + 2x – 3 – 3x2 – 8x + 4
Add theopposite.
Associativeproperty.
= 2x2 – 6x + 1 Combine liketerms.
14-3 Adding Polynomials
Subtract.
Example 1: Subtracting Polynomials Horizontally
B. (b2 + 4b – 1) – (7b2 – b – 1)
= (b2 + 4b – 1) + (–7b2 + b + 1)
= b2 + 4b – 1 – 7b2 + b + 1
= –6b2 + 5b
Add the opposite.
Associative property.
Combine like terms.
14-3 Adding Polynomials
Example 2A
Subtract.
(2y3 + 3y + 5) – (4y3 + 3y + 5)
Add theopposite.
Associativeproperty.
= –2y3 Combine liketerms.
= (2y3 + 3y + 5) + (–4y3 – 3y – 5)
= 2y3 + 3y + 5 – 4y3 – 3y – 5
14-3 Adding Polynomials
Example 2B
Subtract.
(c3 + 2c2 + 3) – (4c3 – c2 – 1)
= –3c3 + 3c2 + 4
Add the opposite.
Associative property.
Combine like terms.
= (c3 + 2c2 + 3) + (–4c3 + c2 + 1)
= c3 + 2c2 + 3 – 4c3 + c2 + 1
14-3 Adding Polynomials
You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms.
14-3 Adding Polynomials
Subtract.
Example 3: Subtracting Polynomials Vertically
(2n2 – 4n + 9) – (6n2 – 7n + 5)
(2n2 – 4n + 9) 2n2 – 4n + 9
– (6n2 – 7n + 5) + –6n2 + 7n – 5
–4n2 + 3n + 4
Add the opposite.
14-3 Adding Polynomials
Subtract.
Example 4: Subtracting Polynomials Vertically
(10x2 + 2x – 7) – (x2 + 5x + 1)
(10x2 + 2x – 7) 10x2 + 2x – 7
– (x2 + 5x + 1) + –x2 – 5x – 1
9x2 – 3x – 8
Add the opposite.
14-3 Adding Polynomials
Subtract.
Example 5: Subtracting Polynomials Vertically
(6a4 – 3a2 – 8) – (–2a4 + 7)
(6a4 – 3a2 – 8) 6a4 – 3a2 – 8– (–2a4 + 7) + 2a4 – 7
8a4 – 3a2 – 15
Rearrange as needed.
14-3 Adding Polynomials
Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems
14-3 Adding Polynomials
Lesson Quiz: Part I
Add.
1. (2m2 – 3m + 7) + (7m2 – 1)
2. (yz2 + 5yz + 7) + (2yz2 – yz)
3.
9m2 – 3m + 6
3yz2+ 4yz + 7
7xy + 2x + 3y + 22
(2xy2 + 2x – 6) + (5xy2 + 3y + 8)
14-3 Adding Polynomials
Lesson Quiz
Find the opposite of each polynomial.
Subtract.
3. (3z2 – 7z + 6) – (2z2 + z – 12)
3m3 – 2m2n
z2 – 8z + 18
–3a2b2c3 2. –3m3 + 2m2n
4. –18h3 – (4h3 + h2 – 12h + 2)
5. (3b2c + 5bc2 – 8b2)
– (4b2c + 2bc2 – c2)
– b2c + 3bc2 – 8b2 + c2
– 22h3 – h2 + 12h – 2
1. 3a2b2c3
14-3 Adding Polynomials
1. Add (4p2 – 8p +11) + (6p2 – 9).
A. 10p2 – 8p + 2
B. 10p2 + 8p + 20
C. 2p2 + 8p + 2
D. 10p2 – 8p + 20
Lesson Quiz for Student Response Systems
14-3 Adding Polynomials
2. Add (gh2 + 9gh + 11) + (3gh2 – gh).
A. 2gh2 + 8gh + 11
B. 2gh2 + 10gh + 11
C. 4gh2 + 8gh + 11
D. 4gh2 + 10gh + 11
Lesson Quiz for Student Response Systems
14-3 Adding Polynomials
3. Add (7uv3 + 11u) + (6uv3 – u – 9) + (4u – 2).
A. 13uv3 – 6u – 7
B. uv2 – 16u + 7
C. uv3 + 14u – 11
D. 13uv3 + 14u – 11
Lesson Quiz for Student Response Systems
14-3 Adding Polynomials
3. Subtract.(11p2 – 5p + 9) – (3p2 + p – 17)
A. 14p2 + 4p – 8
B. 14p2 – 6p + 26
C. 8p2 – 6p + 26
D. 8p2 + 4p – 8
Lesson Quiz for Student Response Systems