polynomials

POLYNOMIALSMs. Johnson 8th Grade Math

KEY VOCABULARY Variable- A quantity that can change or vary,

taking on different values

KEY VOCABULARY Variable- A quantity that can change or vary, taking on

different values

Constant- A quantity having a fixed value that does not change or vary, such as a number. A constant term does not contain a variable. [( y= 5 + x ) in this example, 5 is a constant

KEY VOCABULARY Variable- A quantity that can change or vary, taking

on different values


Coefficient- A number that multiplies a variable [ (4b) in this example, 4 is a coefficient]

KEY VOCABULARY Variable- A quantity that can change or vary, taking on

different values


Coefficient- A number that multiplies a variable [ (4b) in this example, 4 is a coefficient]

Like Terms- Terms whose variables and exponents are the same (the coefficients can be different) [5a + 8a ; or x2y + 3x2y]

KEY VOCABULARY Monomial- a polynomial containing one term which

may be a number, a variable, or a product of numbers and variables, with no negative or fractional exponents

( -5, xy, 5x2, 5x2y3z )



( -5, xy, 5x2, 5x2y3z )

Binomial- a polynomial containing two unlike terms ( x+ 5, xy + 5, 5x2 – x, 5x2 + y3)



( -5, xy, 5x2, 5x2y3z )

Binomial- a polynomial containing two unlike terms ( x+ 5, xy +5, 5x2 – x, 5x2 + y3)

Trinomial- a polynomial containing three unlike terms

(x+5+y, 5x2 –x+y, 5x2 + y3 – z)

KEY VOCABULARY Monomial- a polynomial containing one term which may

be a number, a variable, or a product of numbers and variables, with no negative or fractional exponents

( -5, xy, 5x2, 5x2y3z )

Binomial- a polynomial containing two unlike terms ( x+ 5, xy+5, 5x2 – x, 5x2 + y3)

Trinomial- a polynomial containing three unlike terms(x+5+y, 5x2 –x+y, 5x2 + y3 – z)

Polynomial- an expression that is a monomial or the sum of monomials (xy, xy- 5 , 5x2 – x + y, 5x2 + y3 – z +3)

8.A.6 MULTIPLY MONOMIALS

Just remember the Product Law!

12a3b2 (3a4b6)

Product Law2 applies to the coefficients

Product Law1 applies to the variables

121a3b2 (31a4b6) =


Just remember the Product Law!

12a3b2 (3a4b6)

Product Law2 applies to the coefficients

Product Law1 applies to the variables

121a3b2 (31a4b6) = 361a7b8 = 36a7b8


Some examples:

a. 5bc (4b3c2) =*don’t forget about the hidden ones

b. 4a3mr (6am4) =*if a variable only appears once, keep it in the final answer

c. -2b3m4 (3b2m-2) =*don’t forget about integer rules!

d. 12a4bg3 (-2abg3) =

20b4c3

24a4m5r

-6b5m2

-24a5b2g6


Now You Try Some!

a) 4a3m (8a4m6)

b) 5c4a3 (-3c2a)

c) -6m4r-2 (3m0r6)

d) 2b6 (8b3c)


Now You Try Some!

a) 4a3m (8a4m6) = 32a7m7

b) 5c4a3 (-3c2a)

c) -6m4r-2 (3m0r6)

d) 2b6 (8b3c)


Now You Try Some!

a) 4a3m (8a4m6) = 32a7m7

b) 5c4a3 (-3c2a) = -15a4c6

c) -6m4r-2 (3m0r6)

d) 2b6 (8b3c)


Now You Try Some!

a) 4a3m (8a4m6) = 32a7m7

b) 5c4a3 (-3c2a) = -15a4c6

c) -6m4r-2 (3m0r6) = -18m4r4

d) 2b6 (8b3c)


Now You Try Some!

a) 4a3m (8a4m6) = 32a7m7

b) 5c4a3 (-3c2a) = -15a4c6

c) -6m4r-2 (3m0r6) = -18m4r4

d) 2b6 (8b3c) = 16b9c

8.A.6 DIVIDE MONOMIALS

Just remember the quotient laws!

27a5b6 ÷ 9a2b

Quotient Law2 let’s us divide 27 by 9 = 3

Quotient Law1 lets us divide the variables


Just remember the quotient laws!

27a5b6 ÷ 9a2b

Quotient Law2 let’s us divide 27 by 9 = 3

Quotient Law1 lets us divide the variables (which means subtract the exponents for each variable)

= 3a3b5


Some Examples:

a)4a6b3 ÷ 2a3b

a)12m4n2 ÷ 4mn5

a)2a5b3 ÷ 3a3

b)5m7c4 ÷ 5m7c-9


Some Examples:

a)4a6b3 ÷ 2a3b = 2a3b2

a)12m4n2 ÷ 4mn5

a)2a5b3 ÷ 3a3

b)5m7c4 ÷ 5m7c-9


Some Examples:

a)4a6b3 ÷ 2a3b = 2a3b2

a)12m4n2 ÷ 4mn5 = 3m3n-3

a)2a5b3 ÷ 3a3

a)5m7c4 ÷ 5m7c-9


Some Examples:

a)4a6b3 ÷ 2a3b = 2a3b2

a)12m4n2 ÷ 4mn5 = 3m3n-3

a)2a5b3 ÷ 3a3 = (2/3) a2b3

a)5m7c4 ÷ 5m7c-9


Some Examples:

a)4a6b3 ÷ 2a3b = 2a3b2

a)12m4n2 ÷ 4mn5 = 3m3n-3

a)2a5b3 ÷ 3a3 = (2/3) a2b3

5m7c4 ÷ 5m7c-9 = 1m0c13


Some Examples:

a)4a6b3 ÷ 2a3b = 2a3b2

a)12m4n2 ÷ 4mn5 = 3m3n-3

a)2a5b3 ÷ 3a3 = (2/3) a2b3

a)5m7c4 ÷ 5m7c-9 = 1m0c13 = c13


Now You Try Some…Alone!

a) 30a6 ÷ 5a

a) 32b4m3 ÷ 8bm3

a) 16a7c3 ÷ 4a9

9b6r5 ÷ 6b2r-4



a) 30a6 ÷ 5a = 6a5

a) 32b4m3 ÷ 8bm3

a) 16a7c3 ÷ 4a9

9b6r5 ÷ 6b2r-4



a) 30a6 ÷ 5a = 6a5

a) 32b4m3 ÷ 8bm3 = 4b3m0

a) 16a7c3 ÷ 4a9

a) 9b6r5 ÷ 6b2r-4



a) 30a6 ÷ 5a = 6a5

a) 32b4m3 ÷ 8bm3 = 4b3m0 = 4b3

a) 16a7c3 ÷ 4a9

a) 9b6r5 ÷ 6b2r-4



a) 30a6 ÷ 5a = 6a5

a) 32b4m3 ÷ 8bm3 = 4b3m0 = 4b3

a) 16a7c3 ÷ 4a9 = 4a-2c3

a) 9b6r5 ÷ 6b2r-4



a) 30a6 ÷ 5a = 6a5

a) 32b4m3 ÷ 8bm3 = 4b3m0 = 4b3

a) 16a7c3 ÷ 4a9 = 4a-2c3

a) 9b6r5 ÷ 6b2r-4 = (3/2)b4r9

8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL

This works a lot like Dividing a monomial by a monomial

15a4b7 + 6a2b4

3abBreak this into two separate fractions:

8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL This works a lot like Dividing a monomial by a

monomial

15a4b7 + 6a2b4

3ab

Break this into two separate fractions:


15a4b7 + 6a2b4

3ab

Break this into two separate fractions:

15a4b7 = 6a2b4 =3ab 3ab


15a4b7 + 6a2b4

3abNow follow Quotient Laws!

15a4b7 = 5a3b6 6a2b4 = 2ab3

3ab 3ab


15a4b7 = 5a3b6 6a2b4 = 2ab3

3ab 3abPut these two parts together, using the

operation from the original! In this case, it was an addition problem, so the answer is…

5a3b6 + 2ab3


One more example:

A. 18m6n9 − 24m4n7

6m2n3


One more example:

A. 18m6n9 − 24m4n7

6m2n3

*break into two fractions!

18m6n9 = 24m4n7 = 6m2n3 6m2n3


One more example:

A. 18m6n9 − 24m4n7

6m2n3

Follow quotient laws! 18m6n9 = 3m4n6 24m4n7 = 6m2n3 6m2n3


One more example:

A. 18m6n9 − 24m4n7

6m2n3

Follow quotient laws

18m6n9 = 3m4n6 24m4n7 = 4m2n4 6m2n3 6m2n3


One more example:

A. 18m6n9 − 24m4n7

6m2n3

*follow quotient laws!

18m6n9 = 3m4n6 24m4n7 = 4m2n4 6m2n3 6m2n3

*put it all together:

3m4n6 − 4m2n4

8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL Now You Try Alone ! And don’t forget integer

rules!

A.30a6b2 − 27a4b6

3a3b4

B. 28m5n3p4 + 32m2n5p 4mn7p3


Let’s Go Over Them!

A.30a6b2 − 27a4b6

3a3b4

30a6b2 = 27a4b6 = 3a3b4 3a3b4



A.30a6b2 − 27a4b6

3a3b4

30a6b2 = 10a3b-2 27a4b6 = 3a3b4 3a3b4



A.30a6b2 − 27a4b6

3a3b4

30a6b2 = 10a3b-2 27a4b6 = 9ab2

3a3b4 3a3b4


Let’s Go Over Them!A. 30a6b2 − 27a4b6

3a3b4

30a6b2 = 10a3b-2 27a4b6 = 9ab2

3a3b4 3a3b

10a3b-2 − 9ab2


Now You Try! And don’t forget integer rules!

B. 28m5n3p4 + 32m2n5p 4mn7p3



B. 28m5n3p4 + 32m2n5p 4mn7p3

*Break into 2 fractions

28m5n3p4 32m2n5p

4mn7p3 4mn7p3



B. 28m5n3p4 + 32m2n5p 4mn7p3


28m5n3p4 = 7m4n-4p 32m2n5p =4mn7p3 4mn7p3



B. 28m5n3p4 + 32m2n5p 4mn7p3


28m5n3p4 = 7m4n-4p 32m2n5p = 8mn-2p-2

4mn7p3 4mn7p3



B. 28m5n3p4 + 32m2n5p

4mn7p3


28m5n3p4 = 7m4n-4p 32m2n5p = 8mn-2p-2

4mn7p3 4mn7p3

7m4n-4p + 8mn-2p-2

8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL On loose leaf! On your own! Show the broken up

fractions!

1. 36a5b7 + 12a3b4

6ab4

2. 21c8d4 − 35c9d7c8d3

3. 40a6c5 + 60a2c7

20ac4

8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL Let’s go over them…

1. 36a5b7 + 12a3b4 = 6a4b3 + 2a2

6ab4

2. 21c8d4 − 35c9d7c8d3

3. 40a6c5 + 60a2c7

20ac4


1. 36a5b7 + 12a3b4 = 6a4b3 + 2a2

6ab4

2. 21c8d4 − 35c9d = 3d − 5cd-2 7c8d3

3. 40a6c5 + 60a2c7

20ac4


1. 36a5b7 + 12a3b4 = 6a4b3 + 2a2

6ab4

2. 21c8d4 − 35c9d = 3d − 5cd-2 7c8d3

3. 40a6c5 + 60a2c7 = 2a5c + 3ac3

20ac4

8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL

5(4 + 7) =

There are two ways to evaluate this equation. 1. PEMDAS

1. Distributive Property 5(4+7)


5(4 + 7) =

There are two ways to evaluate this equation. 1. PEMDAS Parentheses: 4+7 =11



5(4 + 7) =There are two ways to evaluate this equation. 1. PEMDAS Parentheses: 4+7 =11 Multiplication: 5(11) = 55




1. Distributive Property 5(4+7) = 5(4) +



1. Distributive Property 5(4+7) = 5(4) + 5(7)



1. Distributive Property 5(4+7) = 5(4) + 5(7) = 20 + 35

8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL 5(4 + 7) =There are two ways to evaluate this equation. 1. PEMDAS Parentheses: 4+7 =11 Multiplication: 5(11) = 55

1. Distributive Property 5(4+7) = 5(4) + 5(7) = 20 + 35

= 55


Remember the Distributive Property (the milkman delivers to each house separately):

a(b + c) = ab +ac

5(c − 6d) =

-4a(7 + 6d) =



a(b + c) = ab +ac

5(c − 6d) = 5c

-4a(7 + 6d) =



a(b + c) = ab +ac

5(c − 6d) = 5c − 30dand these are not like terms, so this is the final answer!

-4a(7 + 6d) =



a(b + c) = ab +ac


-4a(7 + 6d) = -28a



a(b + c) = ab +ac


-4a(7 + 6d) = -28a − 24ad


Some Examples:

1.5z(a2 − 3b)

1.4r3(8r + 7)

1.6b2(7b + 3b)


Some Examples:

1.5z(a2 − 3b) = 5za2 − 15zb

1.4r3(8r + 7)

1.6b2(7b + 3b)


Some Examples:

1.5z(a2 − 3b) = 5za2 − 15zb

2.4r3(8r + 7) = 32r4 + 28r3 Don’t forget about product law! These aren’t like terms!!

1.6b2(7b + 3b)


Some Examples:

1.5z(a2 − 3b) = 5za2 − 15zb


1.6b2(7b + 3b)= 42b3 + 18b3


Some Examples:

1.5z(a2 − 3b) = 5za2 − 15zb


1.6b2(7b + 3b)= 42b3 + 18b3 = 60b3

These are like terms!


Now you try some… ALONE!

1. -4c(7 − 6d)

2. 8m(6m4 + m5)

3. -7a(4 − a)

4. 9b3 (7ab − 4ab4)



1. -4c(7 − 6d) = -28c + 24cd

2. 8m(6m4 + m5)

3. -7a(4 − a)

4. 9b3 (7ab − 4ab4)



1. -4c(7 − 6d) = -28c + 24cd *NB: a negative times a negative is a positive

1. 8m(6m4 + m5)

2. -7a(4 − a)

3. 9b3 (7ab − 4ab4)




8m(6m4 + m5) = 48m5 + 8m6

-7a(4 − a)

9b3 (7ab − 4ab4)




1. 8m(6m4 + m5) = 48m5 + 8m6

2. -7a(4 − a) = -28a +7a2

3. 9b3 (7ab − 4ab4)




1. 8m(6m4 + m5) = 48m5 + 8m6

2. -7a(4 − a) = -28a +7a2

3. 9b3 (7ab − 4ab4) = 63ab4 − 36ab7

8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL

(n+4) (n−5)

For problems like this, we will learn about an important acronym : FOIL

Basically you use the distributive property twice.


(n+4) (n−5)


(n+4) (n−5)

F- firsts n n = n2


(n+4) (n−5)

F- firsts n n = n2

O- outers n -5 = -5n


(n+4) (n−5)

F- firsts n n = n2

O- outers n -5 = -5nI- inners 4 n = 4n


(n+4) (n−5)

F- firsts n n = n2

O- outers n -5 = -5nI- inners 4 n = 4nL- lasts 4 -5 = -20


(n+4) (n−5)

F- firsts n n = n2

O- outers n -5 = -5n like terms

I- inners 4 n = 4n = -nL- lasts 4 -5 = -20

8.A.7 ADD AND SUBTRACT POLYNOMIALS

(n+4) (n−5)

F- firsts n n = n2

O- outers n -5 = -5n like termsI- inners 4 n = 4n = -nL- lasts 4 -5 = -20

So final answer: n2 − n − 20(the signs become the operators!)


(a+2) (a+6)


(a+2) (a+6)

F- firsts a a= a2


(a+2) (a+6)

F- firsts a a = a2

O- outers a 6 = 6a


(a+2) (a+6)

F- firsts a a = a2

O- outers a 6 = 6aI- inners 2 a = 2a


(a+2) (a+6)

F- firsts a a = a2

O- outers a 6 = 6aI- inners 2 a = 2aL- lasts 2 6 = 12


(a+2) (a+6)

F- firsts a a = a2

O- outers a 6 = 6a like terms

I- inners 2 a = 2a = 8aL- lasts 2 6 = 12


(a+2) (a+6)

F- firsts a a = a2

O- outers a 6 = 6a like termsI- inners 2 a = 2a = 8aL- lasts 2 6 = 12

So final answer: a2 + 8a + 12(the signs become the operators!)


1. (a+3)(a−5)

2. (b−4)(b−3)

3. (c+6)(c−4)

4. (d−5)(d+5)

5. (3e+4)(2e−2)


1. (a+3)(a−5) = a2 − 2a − 15

2. (b−4)(b−3)

3. (c+6)(c−4)

4. (d−5)(d+5)

5. (3e+4)(2e−2)


1. (a+3)(a−5) = a2 − 2a − 15

2. (b−4)(b−3) = b2 − 7b + 12

3. (c+6)(c−4)

4. (d−5)(d+5)

5. (3e+4)(2e−2)


1. (a+3)(a−5) = a2 − 2a − 15

2. (b−4)(b−3) = b2 − 7b + 12

3. (c+6)(c−4) = c2 + 2c − 24

4. (d−5)(d+5)

5. (3e+4)(2e−2)


1. (a+3)(a−5) = a2 − 2a − 15

2. (b−4)(b−3) = b2 − 7b + 12

3. (c+6)(c−4) = c2 + 2c − 24

4. (d−5)(d+5) = d2 − 25

5. (3e+4)(2e−2)


1. (a+3)(a−5) = a2 − 2a − 15

2. (b−4)(b−3) = b2 − 7b + 12

3. (c+6)(c−4) = c2 + 2c − 24

4. (d−5)(d+5) = d2 − 25

5. (3e+4)(2e−2)= 6e2 +2e −8


Different Method- Area Model (n + 7)(n+3)

n + 7

n

+ 3



n + 7

n

+ 3

n2



n + 7

n

+ 3

n2 + 7n



n + 7

n

+ 3

n2 + 7n

+3n



n + 7

n

+ 3

n2 + 7n

+3n +21



n + 7

n

+ 3

n2 + 10n + 21

n2 + 7n

+3n +21


(a-5)(a+4) a -5

a

+ 4


(a-5)(a+4) a -5

a

+ 4

a2


(a-5)(a+4) a -5

a

+ 4

a2 -5a


(a-5)(a+4) a -5

a

+ 4

a2 -5a

+4a


(a-5)(a+4) a -5

a

+ 4

a2 -5a

+4a -20


(a-5)(a+4) a -5

a

+ 4 a2 – 1a – 20

a2 -5a

+4a -20


(a – 6)(a + 5)

(b + 4)(b – 8)

(c – 3)(4c + 5)

(3d – 6)(4d + 2)


(a – 6)(a + 5) = a2 – 1a – 30

(b + 4)(b – 8)

(c – 3)(4c + 5)

(3d – 6)(4d + 2)


(a – 6)(a + 5) = a2 – 1a – 30

(b + 4)(b – 8) = b2 – 4b – 32

(c – 3)(4c + 5)

(3d – 6)(4d + 2)


(a – 6)(a + 5) = a2 – 1a – 30

(b + 4)(b – 8) = b2 – 4b – 32

(c – 3)(4c + 5) = 4c2 – 7c – 15

(3d – 6)(4d + 2)


(a – 6)(a + 5) = a2 – 1a – 30

(b + 4)(b – 8) = b2 – 4b – 32

(c – 3)(4c + 5) = 4c2 – 7c – 15

(3d – 6)(4d + 2) = 12d2 – 18d – 12


The main concept is Like Terms: matching variables (including their exponents)

LIKE: can be combined! Rules of exponents don’t come into play because you’re not x/÷. Leave variables as is!!

ax + 5ax

NON-LIKE




ax + 5ax = 6ax 2b2 + 6b2

NON-LIKE




ax + 5ax = 6ax 2b2 + 6b2 = 8b2

NON-LIKE




ax + 5ax = 6ax 2b2 + 6b2 = 8b2

-3acd − 4acd

NON-LIKE




ax + 5ax = 6ax 2b2 + 6b2 = 8b2

-3acd − 4acd = -7acd

NON-LIKE

8.A.7 ADD AND SUBTRACT POLYNOMIALS The main concept is Like Terms: matching

variables (including their exponents) LIKE: can be combined! Rules of exponents don’t

come into play because you’re not x/÷. Leave variables as is!!

ax + 5ax = 6ax 2b2 + 6b2 = 8b2


NON-LIKE: can’t be combined!

8.A.7 ADD AND SUBTRACT POLYNOMIALS The main concept is Like Terms: matching variables

(including their exponents) LIKE: can be combined! Rules of exponents don’t


ax + 5ax = 6ax 2b2 + 6b2 = 8b2


NON-LIKE: can’t be combined!5x – 4y

8.A.7 ADD AND SUBTRACT POLYNOMIALS The main concept is Like Terms: matching variables

(including their exponents) LIKE: can be combined! Rules of exponents don’t


ax + 5ax = 6ax 2b2 + 6b2 = 8b2


NON-LIKE: can’t be combined!5x – 4y 4a3 + 3a

COMBINING LIKE TERMS

Always simplify to the minimum number of terms

Constants can be combined



Use the sign in front of the term as the operator Constants can be combined Examples:

1) 5x – 6 + 2x =



Use the sign in front of the term as the operator Constants can be combined Examples:

1) 5x – 6 + 2x = 7x – 6

COMBINING LIKE TERMS Always simplify to the minimum number of

terms Use the sign in front of the term as the operator Constants can be combined Examples:

1) 5x – 6 + 2x = 7x – 6

2) -8 + 4g – 5 + 2g=



1) 5x – 6 + 2x = 7x – 6

2) -8 + 4g – 5 + 2g= 6g – 13



1) 5x – 6 + 2x = 7x – 6

2) -8 + 4g – 5 + 2g= 6g – 13

3) 3(5 + m) + 4 – 5m=



1) 5x – 6 + 2x = 7x – 6

2) -8 + 4g – 5 + 2g= 6g – 13

3) 3(5 + m) + 4 – 5m= 15 + 3m + 4 – 5m =



1) 5x – 6 + 2x = 7x – 6

2) -8 + 4g – 5 + 2g= 6g – 13

3) 3(5 + m) + 4 – 5m= 15 + 3m + 4 – 5m = 19 – 2m


You Try Some: 1) 5a + 8 – 7a + 2

2) 3p – 3p + 2

3) 5h + 4h + 7h – 16h

4) 7 – 4a + 3a – 10

5) 4 – 2(b – 4) + 4b


You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2

3) 5h + 4h + 7h – 16h

4) 7 – 4a + 3a – 10

5) 4 – 2(b – 4) + 4b


You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2 = 0p + 2

3) 5h + 4h + 7h – 16h

4) 7 – 4a + 3a – 10

5) 4 – 2(b – 4) + 4b


You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2 = 0p + 2 = 2

3) 5h + 4h + 7h – 16h

4) 7 – 4a + 3a – 10

5) 4 – 2(b – 4) + 4b


You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2 = 0p + 2 = 2

3) 5h + 4h + 7h – 16h = 0h = 0

4) 7 – 4a + 3a – 10

5) 4 – 2(b – 4) + 4b


You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2 = 0p + 2 = 2

3) 5h + 4h + 7h – 16h = 0h = 0

4) 7 – 4a + 3a – 10 = -3 – 1a

5) 4 – 2(b – 4) + 4b

COMBINING LIKE TERMS You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2 = 0p + 2 = 2

3) 5h + 4h + 7h – 16h = 0h = 0

4) 7 – 4a + 3a – 10 = -3 – 1a

5) 4 – 2(b – 4) + 4b = 4 – 2b + 8 + 4b =

COMBINING LIKE TERMS You Try Some: 1) 5a + 8 – 7a + 2 = -2a + 10

2) 3p – 3p + 2 = 0p + 2 = 2

3) 5h + 4h + 7h – 16h = 0h = 0

4) 7 – 4a + 3a – 10 = -3 – 1a

5) 4 – 2(b – 4) + 4b = 4 – 2b + 8 + 4b = 12 + 2b

8.A.7 ADD POLYNOMIALS

Line up like terms vertically!!

(8a4 + 2a6) + (5a4 + 3a6)

(6b3 + 2b) + (5b − 4b3)



(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6

(6b3 + 2b) + (5b − 4b3)



(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6

13a4 + 5a6

(6b3 + 2b) + (5b − 4b3)



(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6

13a4 + 5a6 This can’t be simplified further

(6b3 + 2b) + (5b − 4b3)



(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6


(6b3 + 2b) + (5b − 4b3) −4b3 + 5b



(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6


(6b3 + 2b) + (5b − 4b3) −4b3 + 5b It switches to line up like terms



(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6


(6b3 + 2b) + (5b − 4b3) −4b3 + 5b It switches to line up like terms

2b3 + 7b



(8c5 + 3c) + (-5c5 + 6c2)



(8c5 + 3c) + (-5c5 + 6c2)

-5c5 + 6c2



(8c5 + 3c) + (-5c5 + 6c2)

–5c5 + 6c2



(8c5 + 3c) + (-5c5 + 6c2)

–5c5 + 6c2

3c5 + 3c + 6c2



(8c5 + 3c) + (-5c5 + 6c2)

–5c5 + 6c2 So if there’s no like term, leave 3c5 + 3c + 6c2 a gap


You Try!

1. (8a3 + 1a) + (6a + 7a3)

2. (6x2 + 11yz) + (8x2 – 9yz)

3. (3b4 – 2b2) + (6b4 + ba + 9b2)


You Try!

1. (8a3 + 1a) + (6a + 7a3)15a3 + 7a

2. (6x2 + 11yz) + (8x2 – 9yz)

3. (3b4 – 2b2) + (6b4 + ba + 9b2)


You Try!

1. (8a3 + 1a) + (6a + 7a3)15a3 + 7a

2. (6x2 + 11yz) + (8x2 – 9yz)14x2 + 2yz

3. (3b4 – 2b2) + (6b4 + ba + 9b2)


You Try!

1. (8a3 + 1a) + (6a + 7a3)15a3 + 7a

2. (6x2 + 11yz) + (8x2 – 9yz)14x2 + 2yz

3. (3b4 – 2b2) + (6b4 + ba + 9b2)9b4 + 7b2 + ba


Worksheet!



(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3

7c4 + 6c + 5c3 be careful to line up like terms only!

(d3 – 4d2 + 5) + (2d3 +6d2 + 8d)



(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3


(d3 – 4d2 + 5) + (2d3 +6d2 + 8d) +2d3 +6d2 +8d)



(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3


(d3 – 4d2 + 5) + (2d3 +6d2 + 8d) +2d3 +6d2 +8d) 3d3 + 2d2 +5 + 8d


You Try!1. (4x2y + 3y) + (8x2y +9y)

2. (5a3 + 10a) + (-4a3 + 6a)

3. (2b4 – 7b2) + (5b4 + 12ba + 9b2)

8.A.7 SUBTRACT POLYNOMIALS

(8a2b + 5a) – (5a2b + 2a)Remember! K C CThe operator changes to addition and then all the signs of the

terms on the inside change !

(8a2b + 5a) – (5a2b + 2a)




(8a2b + 5a) – (5a2b + 2a) +




(8a2b + 5a) – (5a2b + 2a) + (-5a2b




(8a2b + 5a) – (5a2b + 2a) + (-5a2b – 2a)




(8a2b + 5a) – (5a2b + 2a) + (-5a2b – 2a) + (-5a2b – 2a)




(8a2b + 5a) – (5a2b + 2a) + (-5a2b – 2a) + (-5a2b – 2a) 3a2b




(8a2b + 5a) – (5a2b + 2a) + (-5a2b – 2a) + (-5a2b – 2a) 3a2b + 3a

8.A.7 SUBTRACT POLYNOMIALS Another example!

(6b3 – 8bc) – (8b3 + 9bc)


(6b3 – 8bc) – (8b3 + 9bc) K C C


(6b3 – 8bc) – (8b3 + 9bc) K C C

+


(6b3 – 8bc) – (8b3 + 9bc) K C C

+ (-8b3


(6b3 – 8bc) – (8b3 + 9bc) K C C

+ (-8b3 – 9bc)


(6b3 – 8bc) – (8b3 + 9bc) K C C

+ (-8b3 – 9bc)+ (-8b3 – 9bc)


(6b3 – 8bc) – (8b3 + 9bc) K C C

+ (-8b3 – 9bc)+ (-8b3 – 9bc) -2b3


(6b3 – 8bc) – (8b3 + 9bc) K C C

+ (-8b3 – 9bc)+ (-8b3 – 9bc) -2b3 – 17bc


(15c3m – 4c2 + 8m) – (9c3m – 10m)


(15c3m – 4c2 + 8m) – (9c3m – 10m) K C C + (-9c3m +10m)


(15c3m – 4c2 + 8m) – (9c3m – 10m) K C C + (-9c3m +10m)

+ (-9c3m +10m) Remember to line up like terms!

6c3m – 4c2 + 18m

8.A.7 SUBTRACT POLYNOMIALS You Try… On your own!(-5c3 – 4c4) – (2c3 + 5c4)

(5a3 + 7b) – (6b – 4a3)

(20v2 – 5m) – (12v2 – 4m + 2)

(8g4 + 6a – 7) – (5g4 – 10)

(9a2 – 5m) – (-6a2 + 8m)

8.A.7 SUBTRACT POLYNOMIALS You Try… On your own!(-5c3 – 4c4) – (2c3 + 5c4)

-7c3 – 9c4

(5a3 + 7b) – (6b – 4a3)

(20v2 – 5m) – (12v2 – 4m + 2)

(8g4 + 6a – 7) – (5g4 – 10)

(9a2 – 5m) – (-6a2 + 8m)

8.A.7 SUBTRACT POLYNOMIALS You Try… On your own! (-5c3 – 4c4) – (2c3 + 5c4)

-7c3 – 9c4

(5a3 + 7b) – (6b – 4a3)9a3 + b

(20v2 – 5m) – (12v2 – 4m + 2)

(8g4 + 6a – 7) – (5g4 – 10)

(9a2 – 5m) – (-6a2 + 8m)


-7c3 – 9c4

(5a3 + 7b) – (6b – 4a3)9a3 + b

(20v2 – 5m) – (12v2 – 4m + 2)8v2 – m – 2

(8g4 + 6a – 7) – (5g4 – 10)

(9a2 – 5m) – (-6a2 + 8m)


-7c3 – 9c4

(5a3 + 7b) – (6b – 4a3)9a3 + b

(20v2 – 5m) – (12v2 – 4m + 2)8v2 – m – 2

(8g4 + 6a – 7) – (5g4 – 10) 3g3 + 6a + 3

(9a2 – 5m) – (-6a2 + 8m)


-7c3 – 9c4

(5a3 + 7b) – (6b – 4a3)9a3 + b

(20v2 – 5m) – (12v2 – 4m + 2)8v2 – m – 2

(8g4 + 6a – 7) – (5g4 – 10) 3g3 + 6a + 3

(9a2 – 5m) – (-6a2 + 8m)15a2 – 13m


1. (4a – 6a3) + (8a3 – 3a) 2. (7g + 2g2) – (7g2 + 2g)

3. (3m2 – 2m + 6) – (3m – 2m2)

4. (5c2 + 3c3) + (4c3 + 1)

5. (7x2 + 4) – (5 + 9x2)

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2)


1. (4a – 6a3) + (8a3 – 3a) = a + 2a3

2. (7g + 2g2) – (7g2 + 2g)

3. (3m2 – 2m + 6) – (3m – 2m2)

4. (5c2 + 3c3) + (4c3 + 1)

5. (7x2 + 4) – (5 + 9x2)

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2)


1. (4a – 6a3) + (8a3 – 3a) = a + 2a3

2. (7g + 2g2) – (7g2 + 2g) = 5g – 5g2

3. (3m2 – 2m + 6) – (3m – 2m2)

4. (5c2 + 3c3) + (4c3 + 1)

5. (7x2 + 4) – (5 + 9x2)

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2)


1. (4a – 6a3) + (8a3 – 3a) = a + 2a3

2. (7g + 2g2) – (7g2 + 2g) = 5g – 5g2

3. (3m2 – 2m + 6) – (3m – 2m2) = 5m2 – 5m + 6

4. (5c2 + 3c3) + (4c3 + 1)

5. (7x2 + 4) – (5 + 9x2)

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2)


1. (4a – 6a3) + (8a3 – 3a) = a + 2a3

2. (7g + 2g2) – (7g2 + 2g) = 5g – 5g2

3. (3m2 – 2m + 6) – (3m – 2m2) = 5m2 – 5m + 6

4. (5c2 + 3c3) + (4c3 + 1) = 7c3 + 5c2 + 1

5. (7x2 + 4) – (5 + 9x2)

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2)


1. (4a – 6a3) + (8a3 – 3a) = a + 2a3

2. (7g + 2g2) – (7g2 + 2g) = 5g – 5g2

3. (3m2 – 2m + 6) – (3m – 2m2) = 5m2 – 5m + 6

4. (5c2 + 3c3) + (4c3 + 1) = 7c3 + 5c2 + 1

5. (7x2 + 4) – (5 + 9x2) = 2x2 – 1

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2)


1. (4a – 6a3) + (8a3 – 3a) = a + 2a3

2. (7g + 2g2) – (7g2 + 2g) = 5g – 5g2

3. (3m2 – 2m + 6) – (3m – 2m2) = 5m2 – 5m + 6

4. (5c2 + 3c3) + (4c3 + 1) = 7c3 + 5c2 + 1

5. (7x2 + 4) – (5 + 9x2) = 2x2 – 1

6. (6n2 – 3n3) – (4n3 – 2n2 + 4n2) = 4n2 – 7n3

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