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Transcript of 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
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
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
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
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 )
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)
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)
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) =
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) = 361a7b8 = 36a7b8
8.A.6 MULTIPLY MONOMIALS
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
8.A.6 MULTIPLY MONOMIALS
Now You Try Some!
a) 4a3m (8a4m6)
b) 5c4a3 (-3c2a)
c) -6m4r-2 (3m0r6)
d) 2b6 (8b3c)
8.A.6 MULTIPLY MONOMIALS
Now You Try Some!
a) 4a3m (8a4m6) = 32a7m7
b) 5c4a3 (-3c2a)
c) -6m4r-2 (3m0r6)
d) 2b6 (8b3c)
8.A.6 MULTIPLY MONOMIALS
Now You Try Some!
a) 4a3m (8a4m6) = 32a7m7
b) 5c4a3 (-3c2a) = -15a4c6
c) -6m4r-2 (3m0r6)
d) 2b6 (8b3c)
8.A.6 MULTIPLY MONOMIALS
Now You Try Some!
a) 4a3m (8a4m6) = 32a7m7
b) 5c4a3 (-3c2a) = -15a4c6
c) -6m4r-2 (3m0r6) = -18m4r4
d) 2b6 (8b3c)
8.A.6 MULTIPLY MONOMIALS
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
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 (which means subtract the exponents for each variable)
= 3a3b5
8.A.6 DIVIDE MONOMIALS
Some Examples:
a)4a6b3 ÷ 2a3b
a)12m4n2 ÷ 4mn5
a)2a5b3 ÷ 3a3
b)5m7c4 ÷ 5m7c-9
8.A.6 DIVIDE MONOMIALS
Some Examples:
a)4a6b3 ÷ 2a3b = 2a3b2
a)12m4n2 ÷ 4mn5
a)2a5b3 ÷ 3a3
b)5m7c4 ÷ 5m7c-9
8.A.6 DIVIDE MONOMIALS
Some Examples:
a)4a6b3 ÷ 2a3b = 2a3b2
a)12m4n2 ÷ 4mn5 = 3m3n-3
a)2a5b3 ÷ 3a3
a)5m7c4 ÷ 5m7c-9
8.A.6 DIVIDE MONOMIALS
Some Examples:
a)4a6b3 ÷ 2a3b = 2a3b2
a)12m4n2 ÷ 4mn5 = 3m3n-3
a)2a5b3 ÷ 3a3 = (2/3) a2b3
a)5m7c4 ÷ 5m7c-9
8.A.6 DIVIDE MONOMIALS
Some Examples:
a)4a6b3 ÷ 2a3b = 2a3b2
a)12m4n2 ÷ 4mn5 = 3m3n-3
a)2a5b3 ÷ 3a3 = (2/3) a2b3
5m7c4 ÷ 5m7c-9 = 1m0c13
8.A.6 DIVIDE MONOMIALS
Some Examples:
a)4a6b3 ÷ 2a3b = 2a3b2
a)12m4n2 ÷ 4mn5 = 3m3n-3
a)2a5b3 ÷ 3a3 = (2/3) a2b3
a)5m7c4 ÷ 5m7c-9 = 1m0c13 = c13
8.A.6 DIVIDE MONOMIALS
Now You Try Some…Alone!
a) 30a6 ÷ 5a
a) 32b4m3 ÷ 8bm3
a) 16a7c3 ÷ 4a9
9b6r5 ÷ 6b2r-4
8.A.6 DIVIDE MONOMIALS
Now You Try Some…Alone!
a) 30a6 ÷ 5a = 6a5
a) 32b4m3 ÷ 8bm3
a) 16a7c3 ÷ 4a9
9b6r5 ÷ 6b2r-4
8.A.6 DIVIDE MONOMIALS
Now You Try Some…Alone!
a) 30a6 ÷ 5a = 6a5
a) 32b4m3 ÷ 8bm3 = 4b3m0
a) 16a7c3 ÷ 4a9
a) 9b6r5 ÷ 6b2r-4
8.A.6 DIVIDE MONOMIALS
Now You Try Some…Alone!
a) 30a6 ÷ 5a = 6a5
a) 32b4m3 ÷ 8bm3 = 4b3m0 = 4b3
a) 16a7c3 ÷ 4a9
a) 9b6r5 ÷ 6b2r-4
8.A.6 DIVIDE MONOMIALS
Now You Try Some…Alone!
a) 30a6 ÷ 5a = 6a5
a) 32b4m3 ÷ 8bm3 = 4b3m0 = 4b3
a) 16a7c3 ÷ 4a9 = 4a-2c3
a) 9b6r5 ÷ 6b2r-4
8.A.6 DIVIDE MONOMIALS
Now You Try Some…Alone!
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:
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:
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
15a4b7 + 6a2b4
3ab
Break this into two separate fractions:
15a4b7 = 6a2b4 =3ab 3ab
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
15a4b7 + 6a2b4
3abNow follow Quotient Laws!
15a4b7 = 5a3b6 6a2b4 = 2ab3
3ab 3ab
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
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
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
One more example:
A. 18m6n9 − 24m4n7
6m2n3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
One more example:
A. 18m6n9 − 24m4n7
6m2n3
*break into two fractions!
18m6n9 = 24m4n7 = 6m2n3 6m2n3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
One more example:
A. 18m6n9 − 24m4n7
6m2n3
Follow quotient laws! 18m6n9 = 3m4n6 24m4n7 = 6m2n3 6m2n3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
One more example:
A. 18m6n9 − 24m4n7
6m2n3
Follow quotient laws
18m6n9 = 3m4n6 24m4n7 = 4m2n4 6m2n3 6m2n3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
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
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
A.30a6b2 − 27a4b6
3a3b4
30a6b2 = 27a4b6 = 3a3b4 3a3b4
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
A.30a6b2 − 27a4b6
3a3b4
30a6b2 = 10a3b-2 27a4b6 = 3a3b4 3a3b4
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
A.30a6b2 − 27a4b6
3a3b4
30a6b2 = 10a3b-2 27a4b6 = 9ab2
3a3b4 3a3b4
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!A. 30a6b2 − 27a4b6
3a3b4
30a6b2 = 10a3b-2 27a4b6 = 9ab2
3a3b4 3a3b
10a3b-2 − 9ab2
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Now You Try! And don’t forget integer rules!
B. 28m5n3p4 + 32m2n5p 4mn7p3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
B. 28m5n3p4 + 32m2n5p 4mn7p3
*Break into 2 fractions
28m5n3p4 32m2n5p
4mn7p3 4mn7p3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
B. 28m5n3p4 + 32m2n5p 4mn7p3
*Break into 2 fractions
28m5n3p4 = 7m4n-4p 32m2n5p =4mn7p3 4mn7p3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
B. 28m5n3p4 + 32m2n5p 4mn7p3
*Break into 2 fractions
28m5n3p4 = 7m4n-4p 32m2n5p = 8mn-2p-2
4mn7p3 4mn7p3
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL
Let’s Go Over Them!
B. 28m5n3p4 + 32m2n5p
4mn7p3
*Break into 2 fractions
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
8.A.9- DIVIDE A POLYNOMIAL BY A MONOMIAL Let’s go over them…
1. 36a5b7 + 12a3b4 = 6a4b3 + 2a2
6ab4
2. 21c8d4 − 35c9d = 3d − 5cd-2 7c8d3
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 − 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)
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
1. Distributive Property 5(4+7)
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)
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) +
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)
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
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
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Remember the Distributive Property (the milkman delivers to each house separately):
a(b + c) = ab +ac
5(c − 6d) =
-4a(7 + 6d) =
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Remember the Distributive Property (the milkman delivers to each house separately):
a(b + c) = ab +ac
5(c − 6d) = 5c
-4a(7 + 6d) =
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Remember the Distributive Property (the milkman delivers to each house separately):
a(b + c) = ab +ac
5(c − 6d) = 5c − 30dand these are not like terms, so this is the final answer!
-4a(7 + 6d) =
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Remember the Distributive Property (the milkman delivers to each house separately):
a(b + c) = ab +ac
5(c − 6d) = 5c − 30dand these are not like terms, so this is the final answer!
-4a(7 + 6d) = -28a
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Remember the Distributive Property (the milkman delivers to each house separately):
a(b + c) = ab +ac
5(c − 6d) = 5c − 30dand these are not like terms, so this is the final answer!
-4a(7 + 6d) = -28a − 24ad
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Some Examples:
1.5z(a2 − 3b)
1.4r3(8r + 7)
1.6b2(7b + 3b)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Some Examples:
1.5z(a2 − 3b) = 5za2 − 15zb
1.4r3(8r + 7)
1.6b2(7b + 3b)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
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)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
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)= 42b3 + 18b3
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
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)= 42b3 + 18b3 = 60b3
These are like terms!
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Now you try some… ALONE!
1. -4c(7 − 6d)
2. 8m(6m4 + m5)
3. -7a(4 − a)
4. 9b3 (7ab − 4ab4)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Now you try some… ALONE!
1. -4c(7 − 6d) = -28c + 24cd
2. 8m(6m4 + m5)
3. -7a(4 − a)
4. 9b3 (7ab − 4ab4)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Now you try some… ALONE!
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)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Now you try some… ALONE!
1. -4c(7 − 6d) = -28c + 24cd *NB: a negative times a negative is a positive
8m(6m4 + m5) = 48m5 + 8m6
-7a(4 − a)
9b3 (7ab − 4ab4)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Now you try some… ALONE!
1. -4c(7 − 6d) = -28c + 24cd *NB: a negative times a negative is a positive
1. 8m(6m4 + m5) = 48m5 + 8m6
2. -7a(4 − a) = -28a +7a2
3. 9b3 (7ab − 4ab4)
8.A.8- MULTIPLY A BINOMIAL BY A MONOMIAL
Now you try some… ALONE!
1. -4c(7 − 6d) = -28c + 24cd *NB: a negative times a negative is a positive
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.
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(n+4) (n−5)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(n+4) (n−5)
F- firsts n n = n2
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(n+4) (n−5)
F- firsts n n = n2
O- outers n -5 = -5n
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(n+4) (n−5)
F- firsts n n = n2
O- outers n -5 = -5nI- inners 4 n = 4n
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(n+4) (n−5)
F- firsts n n = n2
O- outers n -5 = -5nI- inners 4 n = 4nL- lasts 4 -5 = -20
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(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!)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a+2) (a+6)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a+2) (a+6)
F- firsts a a= a2
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a+2) (a+6)
F- firsts a a = a2
O- outers a 6 = 6a
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a+2) (a+6)
F- firsts a a = a2
O- outers a 6 = 6aI- inners 2 a = 2a
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a+2) (a+6)
F- firsts a a = a2
O- outers a 6 = 6aI- inners 2 a = 2aL- lasts 2 6 = 12
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(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
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(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!)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
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)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
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)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
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)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
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)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
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)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
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
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2 + 7n
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2 + 7n
+3n
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2 + 7n
+3n +21
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2 + 7n
+3n +21
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2 + 7n
+3n +21
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
Different Method- Area Model (n + 7)(n+3)
n + 7
n
+ 3
n2 + 10n + 21
n2 + 7n
+3n +21
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
a2
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
a2 -5a
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
a2 -5a
+4a
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
a2 -5a
+4a -20
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
a2 -5a
+4a -20
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4
a2 -5a
+4a -20
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a-5)(a+4) a -5
a
+ 4 a2 – 1a – 20
a2 -5a
+4a -20
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a – 6)(a + 5)
(b + 4)(b – 8)
(c – 3)(4c + 5)
(3d – 6)(4d + 2)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a – 6)(a + 5) = a2 – 1a – 30
(b + 4)(b – 8)
(c – 3)(4c + 5)
(3d – 6)(4d + 2)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(a – 6)(a + 5) = a2 – 1a – 30
(b + 4)(b – 8) = b2 – 4b – 32
(c – 3)(4c + 5)
(3d – 6)(4d + 2)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(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)
8.A.8- MULTIPLY A BINOMIAL BY A BINOMIAL
(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
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
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
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
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
-3acd − 4acd
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
-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
-3acd − 4acd = -7acd
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
come into play because you’re not x/÷. Leave variables as is!!
ax + 5ax = 6ax 2b2 + 6b2 = 8b2
-3acd − 4acd = -7acd
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
come into play because you’re not x/÷. Leave variables as is!!
ax + 5ax = 6ax 2b2 + 6b2 = 8b2
-3acd − 4acd = -7acd
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
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 =
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
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=
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= 6g – 13
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= 6g – 13
3) 3(5 + m) + 4 – 5m=
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= 6g – 13
3) 3(5 + m) + 4 – 5m= 15 + 3m + 4 – 5m =
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= 6g – 13
3) 3(5 + m) + 4 – 5m= 15 + 3m + 4 – 5m = 19 – 2m
COMBINING LIKE TERMS
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
COMBINING LIKE TERMS
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
COMBINING LIKE TERMS
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
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
4) 7 – 4a + 3a – 10
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
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
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)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
(6b3 + 2b) + (5b − 4b3)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
(6b3 + 2b) + (5b − 4b3)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
13a4 + 5a6
(6b3 + 2b) + (5b − 4b3)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
13a4 + 5a6 This can’t be simplified further
(6b3 + 2b) + (5b − 4b3)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
13a4 + 5a6 This can’t be simplified further
(6b3 + 2b) + (5b − 4b3)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
13a4 + 5a6 This can’t be simplified further
(6b3 + 2b) + (5b − 4b3) −4b3 + 5b
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
13a4 + 5a6 This can’t be simplified further
(6b3 + 2b) + (5b − 4b3) −4b3 + 5b It switches to line up like terms
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8a4 + 2a6) + (5a4 + 3a6)+ 5a4 + 3a6
13a4 + 5a6 This can’t be simplified further
(6b3 + 2b) + (5b − 4b3) −4b3 + 5b It switches to line up like terms
2b3 + 7b
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
-5c5 + 6c2
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
-5c5 + 6c2
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
–5c5 + 6c2
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
–5c5 + 6c2
3c5 + 3c + 6c2
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(8c5 + 3c) + (-5c5 + 6c2)
–5c5 + 6c2 So if there’s no like term, leave 3c5 + 3c + 6c2 a gap
8.A.7 ADD POLYNOMIALS
You Try!
1. (8a3 + 1a) + (6a + 7a3)
2. (6x2 + 11yz) + (8x2 – 9yz)
3. (3b4 – 2b2) + (6b4 + ba + 9b2)
8.A.7 ADD POLYNOMIALS
You Try!
1. (8a3 + 1a) + (6a + 7a3)15a3 + 7a
2. (6x2 + 11yz) + (8x2 – 9yz)
3. (3b4 – 2b2) + (6b4 + ba + 9b2)
8.A.7 ADD POLYNOMIALS
You Try!
1. (8a3 + 1a) + (6a + 7a3)15a3 + 7a
2. (6x2 + 11yz) + (8x2 – 9yz)14x2 + 2yz
3. (3b4 – 2b2) + (6b4 + ba + 9b2)
8.A.7 ADD POLYNOMIALS
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
8.A.7 ADD POLYNOMIALS
Worksheet!
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3
7c4 + 6c + 5c3 be careful to line up like terms only!
(d3 – 4d2 + 5) + (2d3 +6d2 + 8d)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3
7c4 + 6c + 5c3 be careful to line up like terms only!
(d3 – 4d2 + 5) + (2d3 +6d2 + 8d)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3
7c4 + 6c + 5c3 be careful to line up like terms only!
(d3 – 4d2 + 5) + (2d3 +6d2 + 8d) +2d3 +6d2 +8d)
8.A.7 ADD POLYNOMIALS
Line up like terms vertically!!
(7c4 + 4c) + (5c3 + 2c)+ 2c + 5c3
7c4 + 6c + 5c3 be careful to line up like terms only!
(d3 – 4d2 + 5) + (2d3 +6d2 + 8d) +2d3 +6d2 +8d) 3d3 + 2d2 +5 + 8d
8.A.7 ADD POLYNOMIALS
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)
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) +
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) + (-5a2b
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) + (-5a2b – 2a)
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) + (-5a2b – 2a)
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) + (-5a2b – 2a)
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) + (-5a2b – 2a) + (-5a2b – 2a)
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) + (-5a2b – 2a) + (-5a2b – 2a) 3a2b
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) + (-5a2b – 2a) + (-5a2b – 2a) 3a2b + 3a
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc)
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3 – 9bc)
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3 – 9bc)
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3 – 9bc)
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3 – 9bc)+ (-8b3 – 9bc)
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3 – 9bc)+ (-8b3 – 9bc) -2b3
8.A.7 SUBTRACT POLYNOMIALS Another example!
(6b3 – 8bc) – (8b3 + 9bc) K C C
+ (-8b3 – 9bc)+ (-8b3 – 9bc) -2b3 – 17bc
8.A.7 SUBTRACT POLYNOMIALS
(15c3m – 4c2 + 8m) – (9c3m – 10m)
8.A.7 SUBTRACT POLYNOMIALS
(15c3m – 4c2 + 8m) – (9c3m – 10m) K C C + (-9c3m +10m)
8.A.7 SUBTRACT POLYNOMIALS
(15c3m – 4c2 + 8m) – (9c3m – 10m) K C C + (-9c3m +10m)
8.A.7 SUBTRACT POLYNOMIALS
(15c3m – 4c2 + 8m) – (9c3m – 10m) K C C + (-9c3m +10m)
8.A.7 SUBTRACT POLYNOMIALS
(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)
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)8v2 – m – 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)8v2 – m – 2
(8g4 + 6a – 7) – (5g4 – 10) 3g3 + 6a + 3
(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)8v2 – m – 2
(8g4 + 6a – 7) – (5g4 – 10) 3g3 + 6a + 3
(9a2 – 5m) – (-6a2 + 8m)15a2 – 13m
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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)
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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)
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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)
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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)
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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)
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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)
8.A.7 ADD AND SUBTRACT POLYNOMIALS
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