Adding and Subtracting Polynomials

1
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
(For help, go to Lesson 1-7.)
Simplify each expression. 1. 6t 13t 2. 5g
34g 3. 7k 15k 4. 2b 6 9b 5. 4n2
7n2 6. 8x2 x2
9-1
2
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
1. 6t 13t (6 13)t 19t 2. 5g 34g (5
34)g 39g 3. 7k 15k (7 15)k 8k 4. 2b
6 9b (2 9)b 6 11b 6 5. 4n2 7n2
(4 7)n2 3n2 6. 8x2 x2 (8 1)x2 7x2
Solutions
9-1
3
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
Find the degree of each monomial.
a. 18
b. 3xy3
c. 6c
9-1
4
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
Write each polynomial in standard form. Then
name each polynomial by its degree and the number
of its terms.
a. 2 7x
linear binomial
b. 3x5 2 2x5 7x
fifth degree trinomial
9-1
5
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
Simplify (6x2 3x 7) (2x2 6x 4).
Method 1 Add vertically.
Line up like terms. Then add the coefficients.
Method 2 Add horizontally.
8x2 3x 3
9-1
6
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
Simplify (2x3 4x2 6) (5x3 2x 2).
Method 1 Subtract vertically.
Line up like terms. Then add the coefficients.
9-1
7
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
(continued)
Method 2 Subtract horizontally.
(2x3 4x2 6) (5x3 2x 2)
2x3 4x2 6 5x3 2x 2 Write the opposite
of each term in the polynomial being subtracted.
9-1
8
Adding and Subtracting Polynomials
ALGEBRA 1 LESSON 9-1
Simplify each expression. Then name each
polynomial by its degree and number of
terms. 1. 4 3x 2x2 2. 2b2 4b3
6 3. (2x4 3x 4) (3x 4 x4) 4. (3r
4r2 3) (4r2 6r 2)
2x2 3x 4 quadratic trinomial
4b3 2b2 6 cubic trinomial
3x4 fourth degree monomial
9r 1 linear binomial
9-1
9
Multiplying and Factoring
ALGEBRA 1 LESSON 9-2
(For help, go to Lesson 17.)
Multiply. 1. 3(302) 2. 41(7) 3. 9(504) Simplify
each expression. 4. 4(6 5x) 5. 8(2y
1) 6. (5v 1)5 7. 7(p 2) 8. (6 x)9 9. 2(4q
1)
9-2
10
Multiplying and Factoring
ALGEBRA 1 LESSON 9-2
1. 3(302) 906 2. 41(7) 287 3. 9(504)
4536 4. 4(6 5x) 4(6) 4(5x) 24
20x 5. 8(2y 1) (8)(2y) (8)(1) 16y
8 6. (5v 1)5 (5v)(5) (1)(5) 25v
5 7. 7(p 2) 7p 7(2) 7p 14 8. (6 x)9
6(9) 9x 54 9x 9. 2(4q 1) (2)(4q)
(2)(1) 8q 2
Solutions
9-2
11
Multiplying and Factoring
ALGEBRA 1 LESSON 9-2
Simplify 2g2(3g3 6g 5).
2g2(3g3 6g 5)
2g2(3g3) 2g2(6g) 2g2(5) Use the
Distributive Property.
6g2 3 12g2 1 10g2 Multiply the
coefficients and add the exponents of powers with
the same base.
6g5 12g3 10g2 Simplify.
9-2
12
Multiplying and Factoring
ALGEBRA 1 LESSON 9-2
Find the GCF of 2x4 10x2 6x.
List the prime factors of each term. Identify the
factors common to all terms.
2x4 2 x x x x 10x2 2 5 x x
6x 2 3 x
The GCF is 2 x, or 2x.
9-2
13
Multiplying and Factoring
ALGEBRA 1 LESSON 9-2
Factor 4x3 12x2 16x.
Step 1 Find the GCF.
4x3 2 2 x x x 12x2 2 2 3 x
x 16x 2 2 2 2 x
4x(x2) 4x(3x) 4x(4)
4x(x2 3x 4)
The GCF is 2 2 x, or 4x.
9-2
14
Multiplying and Factoring
ALGEBRA 1 LESSON 9-2
1. Simplify 2x2(3x2 2x 8). 2. Find the
GCF of 16b4 4b3 8b2. 3. Factor 3x3
9x2. 4. Factor 10y3 5y2 15y.
6x4 4x3 16x2
4b2
3x2(x 3)
5y(2y 3)(y 1)
9-2
15
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
(For help, go to Lesson 9-2.)
Find each product. 1. 4r(r 1) 2. 6h(h2 8h
3) 3. y2(2y3 7) Simplify. Write each answer in
standard form. 4. (x3 3x2 x) (5x2 x
1) 5. (3t3 6t 8) (5t3 7t 2)
6. w(w 1) 4w(w 7) 7. 6b(b 2) b(8b
3) 8. m(4m2 6) 3m2(m 9) 9. 3d2(d3 6)
d3(2d2 4)
9-3
16
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
1. 4r(r 1) 4r(r) 4r(1) 4r 2
4r 2. 6h(h2 8h 3) 6h(h2) 6h(8h) 6h(3)
6h3 48h2 18h 3. y2(2y3 7) y2(2y3) 7y2
2y5 7y2 4. x3 3x2 x 5. 3t3 6t 8
5x2 x 1 5t3 7t 2 x3 8x2
2x 1 8t3 t 6
Solutions
6. w(w 1) 4w(w 7) 7. 6b(b 2) b(8b
3) w(w) w(1) 4w(w) 4w(7) 6b(b)
6b(2) b(8b) b(3) w2 w 4w2 28w 6b2
12b 8b2 3b (1 4)w2 (1 28)w (6
8)b2 (12 3)b 5w2 27w 2b2 15b
9-3
17
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
Solutions (continued)
8. m(4m2 6) 3m2(m 9) m(4m2) m(6)
3m2(m) 3m2(9) 4m3 6m 3m3 27m2 (4
3)m3 27m2 6m 7m3 27m2 6m 9. 3d2(d3
6) d3(2d2 4) 3d2(d3) 3d2(6) d3(2d2)
d3(4) 3d5 18d2 2d5 4d3 (3 2)d5
4d3 18d2 d5 4d3 18d2
9-3
18
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
Simplify (2y 3)(y 2).
(2y 3)(y 2) (2y 3)(y) (2y
3)(2) Distribute 2y 3.
2y2 3y 4y 6 Now distribute y and 2.
2y2 y 6 Simplify.
9-3
19
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
Simplify (4x 2)(3x 6).
The product is 12x2 18x 12.
9-3
20
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
Find the area of the shaded region. Simplify.
area of outer rectangle (3x 2)(2x 1)
area of hole x(x 3)
area of shaded region area of outer rectangle
area of hole
(3x 2)(2x 1) x(x 3) Substitute.
6x2 3x 4x 2 x2 3x Use FOIL to
simplify (3x 2) (2x 1) and the Distributive
Property to simplify x(x 3).
9-3
21
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
Simplify the product (3x2 2x 3)(2x 7).
21x2    14x    21   Multiply by 7.
6x3    4x2     6x Multiply by 2x.
9-3
22
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
(continued)
Method 2 Multiply using the horizontal method.
The product is 6x3 17x2 8x 21.
9-3
23
Multiplying Binomials
ALGEBRA 1 LESSON 9-3
Simplify each product using any method. 1. (x
3)(x 6) 2. (2b 4)(3b 5) 3. (3x 4)(3x2
x 2) 4. Find the area of the shaded region.
x2 3x 18
6b2 22b 20
9x3 9x2 2x 8
2x2 3x 1
9-3
24
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
(For help, go to Lessons 84 and 9-3.)
Simplify. 1. (7x)2 2. (3v)2 3. (4c)2 4. (5g3)2 U
se FOIL to find each product. 5. (j 5)(j
7) 6. (2b 6)(3b 8) 7. (4y 1)(5y
2) 8. (x 3)(x 4) 9. (8c2 2)(c2
10) 10. (6y2 3)(9y2 1)
9-4
25
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
1. (7x)2 72 x2 49x2 2. (3v)2 32 v2
9v2 3. (4c)2 (4)2 c2 16c2 4. (5g3)2 52
(g3)2 25g6
Solutions
5. (j 5)(j 7) (j)(j) (j)(7) (5)(j)
(5)(7) j2 7j 5j 35 j2 12j 35
6. (2b 6)(3b 8) (2b)(3b) (2b)(8)
(6)(3b) (6)(8) 6b2 16b 18b 48
6b2 34b 48
9-4
26
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
Solutions (continued)
7. (4y 1)(5y 2)) (4y)(5y) (4y)(2)
(1)(5y) (1)(2) 20y2 8y 5y 2 20y2
3y 2
8. (x 3)(x 4) (x)(x) (x)(-4) (3)(x)
(3)(4) x2 4x 3x 12 x2 x 12
9. (8c2 2)(c2 10) (8c2)(c2) (8c2)(10)
(2)(c2) (2)(10) 8c4 80c2 2c2 20
8c4 78c2 20
10. (6y2 3)(9y2 1) (6y2)(9y2) (6y2)(1)
(3)(9y2) (3)(1) 54y4 6y2 27y2 3
54y4 21y2 3
9-4
27
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
a. Find (y 11)2.
(y 11)2 y2 2y(11) 72 Square the binomial.
y2 22y 121 Simplify.
b. Find (3w 6)2.
(3w 6)2 (3w)2 2(3w)(6) 62 Square the
binomial.
9w2 36w 36 Simplify.
9-4
28
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
Among guinea pigs, the black fur gene (B) is
dominant and the white fur gene (W) is recessive.
This means that a guinea pig with at least one
dominant gene (BB or BW) will have black fur. A
guinea pig with two recessive genes (WW) will
have white fur.
9-4
29
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
(continued)
9-4
30
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
a. Find 812 using mental math.
812 (80 1)2
b. Find 592 using mental math.
592 (60 1)2
602 2(60 1) 12 Square the binomial.
3600 120 1 3481 Simplify.
9-4
31
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
Find (p4 8)(p4 8).
(p4 8)(p4 8) (p4)2 (8)2 Find the
difference of squares.
p8 64 Simplify.
9-4
32
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
Find 43 37.
43 37 (40 3)(40 3) Express each factor
using 40 and 3.
402 32 Find the difference of squares.
1600 9 1591 Simplify.
9-4
33
Multiplying Special Cases
ALGEBRA 1 LESSON 9-4
Find each square. 1. (y 9)2 2. (2h
7)2 3. 412 4. 292 5. Find (p3 7)(p3
7). 6. Find 32 28.
y2 18y 81
4h2 28h 49
1681
841
p6 49
896
9-4
34
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
(For help, go to Skills Handbook page 721.)
List all of the factors of each
number. 1. 24 2. 12 3. 54 4. 15 5. 36 6. 56 7. 64
8. 96
9-5
35
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
1. Factors of 24 1, 2, 3, 4, 6, 8, 12,
24 2. Factors of 12 1, 2, 3, 4, 6,
12 3. Factors of 54 1, 2, 3, 6, 9, 18, 27,
54 4. Factors of 15 1, 3, 5, 15 5. Factors of
36 1, 2, 3, 4, 6, 9, 12, 18, 36 6. Factors of
56 1, 2, 4, 7, 8, 14, 28, 56 7. Factors of 64
1, 2, 4, 8, 16, 32, 64 8. Factors of 96 1, 2, 3,
4, 6, 8, 12, 16, 24, 32, 48, 96
Solutions
9-5
36
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
Factor x2 8x 15.
Find the factors of 15. Identify the pair that
has a sum of 8.
x2 8x 15 (x 3)(x 5).
9-5
37
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
Factor c2 9c 20.
Since the middle term is negative, find the
negative factors of 20.
Identify the pair that has a sum of 9.
c2 9c 20 (c 5)(c 4)
9-5
38
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
a. Factor x2 13x 48.
b. Factor n2 5n 24.
Identify the pair of factors of 48 that has a
sum of 13.
Identify the pair of factors of 24 that has a
sum of 5.
x2 13x 48 (x 16)(x 3)
n2 5n 24 (n 3)(n 8)
9-5
39
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
Factor d2 17dg 60g2.
d2 17dg 60g2 (d 3g)(d 20g)
9-5
40
Factoring Trinomials of the Type x2 bx c
ALGEBRA 1 LESSON 9-5
Factor each expression. 1. c2 6c 9 2. x2
11x 18 3. g2 2g 24 4. y2 y 110 5. m2
2mn n2
(c 3)(c 3)
(x 2)(x 9)
(g 6)(g 4)
(y 11)(y 10)
(m n)(m n)
9-5
41
Factoring Trinomials of the Type ax2 bx c
ALGEBRA 1 LESSON 9-6
(For help, go to Lessons 9-2 and 9-5.)
Find the greatest common factor. 1. 12x2
6x 2. 28m2 35m 14 3. 4v3 36v2 10 Factor
each expression. 4. x2 5x 4 5. y2 3y
28 6. t2 11t 30
9-6
42
Factoring Trinomials of the Type ax2 bx c
ALGEBRA 1 LESSON 9-6
1. 12x2 6x12x2 2 2 3 x x 6x 2
3 xGCF 2 3 x 6x 2. 28m2 35m
1428m2 2 2 7 m m 35m 5 7 m 14
2 7 GCF 7 3. 4v3 36v2 104v3 2 2
v v v 36v2 2 2 3 3 v v 10
2 5 GCF 2 4. Factors of 4 with a sum of 5 1
and 4x2 5x 4 (x 1)(x 4) 5. Factors of
28 with a sum of 3 4 and 7y2 3y 28 (y
4)(y 7) 6. Factors of 30 with a sum of 11
5 and 6t2 11t 30 (t 5)(t 6)
Solutions
9-6
43
Factoring Trinomials of the Type ax2 bx c
ALGEBRA 1 LESSON 9-6
Factor 20x2 17x 3.
2 10 2 3 1 10 16 1 3 2 1 3 10
32 3 1
9-6
44
Factoring Trinomials of the Type ax2 bx c
ALGEBRA 1 LESSON 9-6
Factor 3n2 7n 6.
3n2 7n 6
(1)(3)     (1)(6) (1)(3) 3 (1)(6)
(1)(1) (6)(3) 17 (6)(1)
(1)(3) (2)(3) 3 (2)(3)
9-6
45
Factoring Trinomials of the Type ax2 bx c
ALGEBRA 1 LESSON 9-6
Factor 18x2 33x 30 completely.
18x2 33x 30 3(6x2 11x 10) Factor out
the GCF.
9-6
46
Factoring Trinomials of the Type ax2 bx c
ALGEBRA 1 LESSON 9-6
Factor each expression. 1. 3x2 14x
11 2. 6t2 13t 63 3. 9y2 48y 36
(x 1)(3x 11)
(2t 9)(3t 7)
3(3y 2)(y 6)
9-6
47
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
(For help, go to Lessons 84 and 9-4.)
Simplify each expression. 1. (3x)2 2. (5y)2 3. (15
h2)2 4. (2ab2)2
Simplify each product. 5. (c 6)(c 6) 6. (p
11)(p 11) 7. (4d 7)(4d 7)
9-7
48
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
Solutions
1. (3x)2 32 x2 9x2 2. (5y)2 52 y2
25y2 3. (15h2)2 152 (h2)2 225h4 4. (2ab2)2
22 a2 (b2)2 4a2b4 5. (c 6)(c 6) is
the difference of squares.(c 6)(c 6) c2
62 c2 36 6. (p 11)(p 11) is the square of
a binomial.(p 11)2 p2 2p(11) 112 p2
22p 121 7. (4d 7)(4d 7) is the square of a
binomial.(4d 7)2 (4d)2 2(4d)(7) 72
16d2 56d 49
9-7
49
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
Factor m2 6m 9.
m2 6m 9 m m 6m 3 3 Rewrite first
and last terms.
m m 2(m 3) 3 3 Does the middle term
equal 2ab? 6m 2(m 3)
(m 3)2 Write the factors as the square of a
binomial.
9-7
50
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
The area of a square is (16h2 40h 25) in.2.
Find the length of a side.
16h2 40h 25 (4h)2 40h 52 Write 16h2 as
(4h)2 and 25 as 52.
(4h 5)2 Write the factors as the square of a
binomial.
The side of the square has a length of (4h 5)
in.
9-7
51
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
Factor a2 16.
a2 16 a2 42 Rewrite 16 as 42.
(a 4)(a 4) Factor.
9-7
52
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
Factor 9b2 25.
9b2 225 (3b)2 52 Rewrite 9b2 as (3b)2 and
25 as 52.
(3b 5)(3b 5) Factor.
9-7
53
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
Factor 5x2 80.
5x2 80 5(x2 16) Factor out the GCF of 5.
5(x 4)(x 4) Factor (x2 16).
9-7
54
Factoring Special Cases
ALGEBRA 1 LESSON 9-7
Factor each expression. 1. y2 18y 81 2. 9a2
24a 16 3. p2 169 4. 36x2 225 5. 5m2
45 6. 2c2 20c 50
(y 9)2
(3a 4)2
(p 13)(p 13)
(6x 15)(6x 15)
5(m 3)(m 3)
2(c 5)2
9-7
55
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
(For help, go to Lessons 9-2 and 9-3.)
Find the GCF of the terms of each
polynomial. 1. 6y2 12y 4 2. 9r3 15r2
21r 3. 30h3 25h2 40h 4. 16m3 12m2
36m Find each product. 5. (v 3)(v2
5) 6. (2q2 4)(q 5) 7. (2t 5)(3t 4) 8. (4x
1)(x2 2x 3)
9-8
56
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
1. 6y2 12y 4 2. 9r3 15r2 21r 6y2 2
3 y y 9r3 3 3 r r r 12y 2
2 3 y 4 2 2 15r2 3 5 r r 21r
3 7 r GCF 2 GCF 3r 3. 30h3 25h2
40h 4. 16m3 12m2 36m 30h3 2 3 5 h
h h 16m3 2 2 2 2 m m m 25h2
5 5 h h 12m2 2 2 3 m m 40h
2 2 2 5 h 36m 2 2 3 3 m GCF
5h GCF 2 2 m 4m
Solutions
5. (v 3)(v2 5) (v)(v2) (v)(5) (3)(v2)
(3)(5) v3 5v 3v2 15 v3 3v2 5v
15
9-8
57
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
Solutions (continued)
6. (2q2 4)(q 5) (2q2)(q) (2q2)(5)
(4)(q) (4)(5) 2q3 10q2 4q 20
7. (2t 5)(3t 4) (2t)(3t) (2t)(4)
(5)(3t) (5)(4) 6t2 8t 15t 20 6t2
7t 20
8. (4x 1)(x2 2x 3) (4x)(x2) (4x)(2x)
(4x)(3) (1)(x2) (1)(2x) (1)(3) 4x3
8x2 12x x2 2x 3 4x3 (8 1)x2
(12 2)x 3 4x3 7x2 10x 3
9-8
58
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
Factor 6x3 3x2 4x 2.
6x3 3x2 4x 2 3x2(2x 1) 2(2x
1) Factor the GCF from each group of two terms.
(2x 1)(3x2 2) Factor out (2x 1).
9-8
59
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
Factor 8t4 12t3 16t 24.
8t4 12t3 16t 24 4(2t4 3t3 4t
6) Factor out the GCF, 4.
4t3(2t 3) 2(2t 3) Factor by grouping.
4(2t 3)(t3 2) Factor again.
9-8
60
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
Factor 24h2 10h 6.
Step 1 24h2 10h 6 2(12h2 5h
3)   Factor out the GCF, 2.
Step 2 12 3 36 Find the product ac.
Step 4  12h2 4h 9h 3 Rewrite the trinomial.
Step 5   4h(3h 1) 3(3h 1) Factor by
grouping.   (4h 3)(3h 1) Factor again.
24h2 10h 6 2(4h 3)(3h 1) Include the
GCF in your final answer.
9-8
61
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
A rectangular prism has a volume of 36x3 51x2
18x. Factor to find the possible expressions
for the length, width, and height of the prism.
Factor 36x3 51x2 18x.
Step 1 3x(12x2 17x 6) Factor out the GCF, 3x.
Step 2 12 6 72 Find the product ac.
9-8
62
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
(continued)
Step 4 3x(12x2 8x 9x 6) Rewrite the
trinomial.
Step 5 3x4x(3x 2) 3(3x 2) Factor by
grouping.
3x(4x 3)(3x 2) Factor again.
The possible dimensions of the prism are 3x, (4x
3), and (3x 2).
9-8
63
Factoring by Grouping
ALGEBRA 1 LESSON 9-8
Factor each expression. 1. 10p3 25p2 4p
10 2. 36x4 48x3 9x2 12x 3. 16a3 24a2
12a 18
(5p2 2)(2p 5)
3x(4x2 1)(3x 4)
2(4a2 3)(2a 3)
9-8