Self-Directed Course: Transitional Math
Module 5: Polynomials
Lesson #1: Properties of Exponents
1) Multiplying Powers with the Same Base
- When multiplying powers that have the same base, add the exponents and keep the base
the same.
- For example:
32 x 33 =
(–5)2 x (–5)1 =
3n4 x 5n3
2+3
2+1
3 =
(–5) =
(3n x 5n)4+3 =
35 =
(–5)3 =
15n4+3 =
243
–125
15n7
2) Dividing Powers with the Same Base
- When dividing powers that have the same base, subtract the exponents and keep the base
the same.
- For example:
46 ÷ 43 =
29 ÷ 24 =
6-3
4 =
29-4 =
43 =
25 =
64
32
3) Raising Powers, Products, and Quotients to an Exponent
- When raising a power to an exponent, multiply the exponents and keep the base the same.
- For example:
(33)2 =
(42)3 =
3x2
3 =
42x3 =
36 =
46 =
729
4096
-
-
When raising a power to a product, you can multiply the product and then to the exponent
or rewrite the product with the same exponent.
For example:
(2 x 4)3 =
(2 x 4)3 =
3
8 =
23 x 43 =
512
8 x 64 =
512
When raising a power to a quotient, rewrite the quotient with the same exponent.
For example:
(3 ÷ 4)3 =
(5 ÷ 3)2 =
33 ÷ 43 =
52 ÷ 32 =
27 ÷ 64 =
25 ÷ 9 =
0.42
2.78
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Assignment #1: Properties of Exponents
Simplify each of the following.
1) 3m2 ● 3m3
18) (x2y1)2
2) m4 ● 3m-3
19) (3x4y3)1
3) 5r3 ● 3r2
20) (4m)2
4) 2n4 ● 5n3
21) 6r12
3r3
5) 3k4 ● 6k
3 3
1 3
6) 3x y ● 3x y
7) 3y2 ● 4x
22) 9x16
3x4
23) 2n4
2n3
8) 3v3 ● vu2
9) 3a3b2 ● 4a4b3
10) x2y4 ● x3y2
24) 14m4
7m4
25) 2m4
2m3
11) (x2)3
12) (4x2)4
13) (3r1)4
26) x4y4z3
x2y3z4
27) 3xy2z3
3x
14) (3a3)2
15) (2k4)4
16) (5xy)1
17) (4b4)1
Property of: Portage la Prairie School Division
28) h3g3k4
gk
29) m4n3p4
m2n2p3
Self-Directed Course: Transitional Math
Module 5: Polynomials
Lesson #2: Combining Expressions
Follow the Properties of Exponents discussed in Lesson #1. Here are a few examples of the type of
questions you will see in Assignment #2. When a variable has no exponent attached to it, it is actually
considered an exponent of one. For example 5n = 5n1.
14x2 – 28x + 35 = 2x2 – 4x + 5
7
3x2 – 4x3 + 3x2
= –4x3 + 6x2
(3xy3z2)(5xyz3)
= 15x2y4z5
3x2 – 3x + x2 – x + 7
= 4x2 – 4x + 7
8x3 – 2x3
= 6x3
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Assignment #2: Combining Expressions
Solve the following expressions.
1) 7x + 5x
15) x2 ● x3
2) 4x2 ● 3x3
16) 6x3 – 2x3
3) x5
x3
17) –5x3 + 2x2 – 3x3
18) 7x + 3x2
4) 6x3 + x3
19) (–5x5)2
5) 3x2 – 4x3 + 3x2
20) 4xy2 + 7xy + 3x2y
6) (3xy3z2)(5xyz3)
7) 3x – 3x + x – x + 7
21) 8x + 4y
4
8) (x3)2
22) x – 4x
9) x2 + 3x2
23) (5xy3)2
(3y2)3
2
2
10) 5x2 ● 2x3 ● 3x2
24) –7x – 4x
4
2 2
11) (4x yz )
12) (–3xy3)3
25) 6x2 – 9x + 18
3
13) (xy2)3 (2x3y4)2
26) (–2x3y)2 (–1x3y4)3
14) (3xy)(4x3y)
27) (4x5y4)(3xy3)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Lesson #3: Adding and Subtracting Polynomials
To add polynomials combine all the like (the same) terms. For example:
(3x – 5) + (5x + 6)
3x + 5x – 5 + 6
8x + 1
3x and 5x have the same variable, so they are the same
Grouping like terms
(3n3 – 5n) + (n3 + 4n + 7)
3n3 + n3 – 5n + 4n + 7
4n3 – 1n + 7
n3 and n are different because the exponents are different
Group like terms
To subtract polynomials add the opposite terms. For example:
(4x – 5) – (2x + 2)
4x – 5 + 2x – 2
4x + 2x – 5 – 2
6x – 7
Re-write with opposite terms
Group like terms
(5n3 – 2m + 5) – (2n3 – 3m – 1)
5n3 – 2m + 5 + 2n3 + 3m + 1
Re-write with opposite terms
3
3
5n + 2n – 2m + 3m + 5 + 1
Group like terms
3
7n + m + 6
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Assignment #3: Adding and Subtracting Polynomials
Solve the following expressions.
1) (5p2 – 3) + (2p2 – 3p3)
7) (5a + 4) – (5a + 3)
2) (a3 – 2a2) – (3a2 – 4a3)
8) (3x4 – 3x) – (3x – 3x4)
3) (4 + 2n3) + (5n3 + 2)
9) (–4k4 + 14 + 3k2) + (–3k4 – 14k2 – 8)
4) (4n – 3n3) – (3n3 + 3n)
10) (3 – 6n5 – 8n4) – (–6n4 – 3n – 8n5)
5) (3a2 + 1) – (4 + 2a2)
11) (12a5 – 6a – 10a3) – (10a – 2a5 – 14a4)
6) (4r3 + 3r4) – (r4 – 5r3)
12) (8n – 3n4 + 10n2) – (3n2 + 11n4 – 7)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
13) (–x4 + 13x5 + 6x3) + (6x3 + 5x5 + 7x4)
20) (9r3 + 5r2 + 11r) + (–2r3 + 9r – 8r2)
14) (13n2 + 11n – 2n4) + (–13n2 – 3n – 6n4)
21) (–7x5 + 14 – 2x) + (10x4 + 7x + 5x5)
15) (7 – 13x3 – 11x) – (2x3 + 8 – 4x5)
22) (13a2 – 6a5 – 2a) – (–10a2 – 11a5 + 9a)
16) (3y5 + 8y3 – 10y2) – (–12y5 + 4y3 + 14y2)
23) (8b3 – 6 + 3b4) – (b4 – 7b3 – 3)
17) (k4 – 3 – 3k3) + (–5k4 + 6k3 – 8k5)
24) (–7n2 + 8n – 4) – (–11n + 2 – 14n2)
18) (–10k2 + 7k + 6k4) + (–14 – 4k4 – 14k)
25) (14p4 + 11p2 – 9p5) – (–14 + 5p5 – 11p2)
19) (–9v2 – 8u) + (–2uv – 2u2 + v2)
26) (8k + k2 – 6) – (–10k + 7 – 2k2)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
27) (4x2 + 7x3y2) – (–6x2 – 7x3y2 – 4x) – (10x + 9x2)
28) (–5u3v4 + 9u) + (–5u3v4 – 8u + 8u2v2) + (–8u4v2 + 8u3v4)
29) (–9xy3 – 9x4y3) + (3xy3 + 7y4 – 8x4y4) + (3x4y3 + 2xy3)
30) (y3 – 7x4y4) + (–10x4y3 + 6y3 + 4x4y4) – (x4y3 + 6x4y4)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Lesson #4: Distributive Property
Monomial x Polynomial
Distributive property allows you to expand an expression by multiplying the first term by each term in
the polynomial. Remember the Properties of Exponents. For example:
3x(5x + 7)
(3x)(5x) + (3x)(7)
(3)(5)(x)(x) + (3)(7)(x)
15x2 + 21x
n(4m – 5n + 3)
(n)(4m) – (n)(5n) + (n)(3)
(4)(m)(n) – (5)(n)(n) + (3)(n)
4mn – 5n2 + 3n
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Assignment #4: Distributive Property
Solve the following expressions.
1) 7(2n + 3m)
11) –9(3b – 2)
2) 4(4r + 5h)
12) –6(j – 1)
3) 6(3w + 2k)
13) –7(r – 4)
4) 5(2q + 4)
14) –(6k – 2)
5) 8(2a + 1)
15) –8(g + h + 2r)
6) 9(2b – 3)
16) –3(4a – 3b – 5c)
7) 3(3m – 4)
17) –2(–w2 + 2w – 5)
8) 4(p – 2)
18) 10(0.4n + 0.2m)
9) –3(8e + 6)
19) 10(0.7n – 2)
10) –5(5q + 2)
20) 100(0.03n + 0.25m)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Lesson #5: Multiplying Polynomials
Polynomial x Polynomial
To multiply two polynomials together, the word FOIL will help you remember each step.
F
O
I
L
first
outer
inner
last
(6n + 3)(3n – 4)
(6n)(3n) = 18n2
(6n)(4) = 24n
(3)(3n) = 9n
(3)( –4) = –12
18n2 + 24n + 9n – 12
18n2 + 33n – 12
(3n2 – 4)(2n2 – 5)
(3n2)(2n2) = 6n4
(3n2)( –5) = –15n2
(–4)(2n2) = –8n2
(–4)( –5) = 20
6n4 – 15n2 – 8n2 + 20
6n4 – 23n2 + 20
first
outer
inner
last
first
outer
inner
last
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
Assignment #5: Multiplying Polynomials
Find the product for the following.
1) 6a(2a + 3)
7) (w – 3)(6w – 2)
2) 7(–5d – 8)
8) (8a – 2)(6a + 2)
3) 2v(–2v – 3)
9) (6q + 8)(5q – 8)
4) –4(g + 1)
10) (3h – 1)(8h + 7)
5) (2r + 2)(6r + 1)
11) (2c – 1)(8c – 5)
6) (4m + 1)(2m + 6)
12) (5k + 6)(5k – 5)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
13) (4n – 1)2
19) (4q + 2)(6q2 – q + 2)
14) (7m – 6)(5m + 6)
20) (7d – 3)(d2 – 2d + 7)
15) (6d + 3)(6d – 4)
21) (7g2 – 6g – 6)(2g – 4)
16) (8f + 1)(6f – 3)
22) (y2 + 6y – 4)(2y – 4)
17) (6w + 5)(5w + 5)
23) (6h2 – 6h – 5)(7h2 + 6h – 5)
18) (3y – 4)(4y + 3)
35) (n2 – 7n – 6)(7n2 – 3n – 7)
24) (y + 5)(y – 2)
25) (g – 1)(g + 1)
Property of: Portage la Prairie School Division
Self-Directed Course: Transitional Math
Module 5: Polynomials
26) (q – 1)2
33) (8m2 + 4)(8m2 – 4)
27) (y – 3)(y + 3)
34) (2 + 5m2)2
28) (y – 4)2
35) (3y – 7)(3y + 7)
29) (m + 3)2
36) (3 + 7x2)(3 – 7x2)
30) (y – 5)(y + 5)
37) (7x2 – 6)(7x2 + 6)
31) (a – 5)2
38) (2 + b)2
32) (2b2 + 1)2
39) (6x + 3)(6x – 3)
Property of: Portage la Prairie School Division