Section P.5 Multiplication of Polynomials
Chapter P – Polynomials
#1-16: Multiply the following monomials
1) 5(4x5)
Just multiply the 5 and the 4, the x5 doesn’t change.
Solution: 20x5
3) (3x2)(-4)
Just multiply the 3 and the -4, the x2 doesn’t change.
Solution: -12x2
5) (3x)(4x3)
Multiply the 3 and 4 and add the exponents of the x’s. Remember the first x has an exponent of 1.
Solution: 12x4
2
6
7) (3 𝑥 2 ) (5 𝑥)
The hard part is the fraction multiplication, cross cancel the 3 with the 6:
2
3
6
5
2
1
2
5
( )( ) = ( )( ) =
After multiplying the fractions, add the exponents of the x’s.
Solution:
𝟒 𝟑
𝒙
𝟓
9) (-a4)(a5)
Think of this as (-1a4)(a5) The -1 will remain, and I just need to add the exponents of the a’s.
Solution: -1a9 which can be written as –a9 (it’s better not to write the 1, but is okay to write it.)
2
7
11) (3 𝑥𝑦 2 ) (4 𝑥𝑦)
Multiply the fractions, you will need to cancel the 2 and the 4.
2
3
7
4
1
3
7
2
∗ = ∗ =
7
6
Then add the exponents of the x and y.
𝟕
Solution: 𝟔 𝒙𝟐 𝒚𝟑
4
5
13) (a2b)(ab3)
= a2a1b1b3
Solution: a3b4
7
15) (−3𝑥) (2 𝑥)
7
First multiply the (−3) (2) this is done by multiplying the -3 and the 7.
7
(−3) ( ) =
2
Solution:
−21
2
, next add the exponents of the x’s.
−21 2
𝑥
2
#17-36: Multiply the following monomial / polynomial products
17) 3x(2x – 5)
= (3x)(2x) + (3x)(-5)
Solution: 6x2 – 15x
19) 7(-2a + 4)
= (7)(-2a) + (7)(4)
Solution: -14a + 28
21) (x – 5)(3x)
= x(3x) + (-5)(3x)
Solution: 3x2 – 15x
23) (-3a + 5)7a
= (-3a)(7a) + 5(7a)
Solution: -21a2 + 35a
25) -2(5x + 8)
= (-2)(5x) + (-2)(8)
Solution: -10x - 16
27) -2x(5x2 + 3x -1)
= (-2x)(5x2) + (-2x)(3x) + (-2x)(-1)
Solution: -10x3 – 6x2 + 2x
2
(4𝑥
3
29)
− 9)
2
2
= (3) (4𝑥) + (3) (−9) I will show the multiplication by the -9 as it is harder. First I make the -9 a
fraction, then I cancel the -9 and the 3
8
2
−9
=3 𝑥 + (3) ( 1 )
8
3
2
1
−3
1
= 𝑥 + ( )( )
8
3
= 𝑥 + (−6)
Solution:
31)
=
𝟖
𝒙−
𝟑
−1
(6𝑥
2
𝟔
− 9)
−1
−1
(6𝑥) + (−9)
2
2
Cancel the 2 and the 6 in the first multiplication, the second multiplication just multiply the (-1) and the
(-9)
𝟗
Solution: −𝟑𝒙 + 𝟐
33) 3y(4y3 – 3y2 + 5y –4)
= (3y)(4y3) + (3y)(-3y2) + (3y)(5y) + (3y)(-4)
Solution: 12y4 – 9y3 + 15y2 – 12y
3
35) 4y(8y3 – 3y2 + 6y –16)
3
4
3
4
3
4
3
4
= ( 𝑦) (8𝑦 3 ) + ( 𝑦) (−3𝑦 2 ) + ( 𝑦) (6𝑦) + ( 𝑦) (−16)
−9
3
= (3𝑦)(2𝑦 3 ) + ( 4 𝑦𝑦 2 ) + (2 𝑦) (3𝑦) + (3𝑦)(−4) I did all the fraction cancelling on this step.
Now I will add the exponents and multiply the numbers that need to be multiplied.
𝟗
𝟗
Solution: 6y4 - 𝟒 𝒚𝟑 + 𝟐 𝒚𝟐 − 𝟏𝟐𝒚
#37 – 52: Multiply the following binomials using FOIL, write the answer in descending order
37) (3x – 4)(2x+5)
Firsts
(3x)(2x)
Outers
(3x)(5)
Inners
(-4)(2x)
lasts
(-4)(5)
= 6x2 + 15x - 8x – 20
Solution: 6x2 + 7x - 20
39) (3x + 1)(-2x –9)
Firsts
(3x)(-2x)
Outers
(3x)(-9)
Inners
(1)(-2x)
lasts
(1)(-9)
Inners
(-2x)(5x)
lasts
(-2x)(3)
= -6x2 – 27x – 2x – 9
Solution: -6x2 – 29x - 9
41) (3 – 2x)(5x + 3)
Firsts
(3)(5x)
Outers
(3)(3)
= 15x + 9 – 10x2 – 6x
Solution: -10x2 + 9x + 9
43) (4 – 9y)(8y + 7)
Firsts
(4)(8y)
Outers
(4)(7)
Inners
(-9y)(8y)
lasts
(-9y)(7)
= 32y + 28 – 72y2 – 63y
Solution: -72y2 – 31y + 28
1
3
45) (2 𝑥 + 6) (2 𝑥 − 5)
Firsts
Outers
Inners
lasts
1
3
1
3
(6)(-5)
( 𝑥) ( 𝑥) ( 𝑥) (−5) (6) ( 𝑥)
2
2
2
2
3
5
= 4 𝑥2 − 2 𝑥 +
Solution:
𝟑 𝟐
𝒙
𝟒
18
𝑥
2
+
− 30
𝟏𝟑
𝒙−
𝟐
𝟑𝟎
3
1
47) (4 𝑦 2 + 5𝑦) (4 𝑦 − 2)
Firsts
Outers
Inners
lasts
3 2 1
3 2
1
(5y)(-2)
( 𝑦 ) ( 𝑦) ( 𝑦 ) (−2) (5𝑦) ( 𝑦)
4
4
4
4
3
3
5
3
6
5
= 16 𝑦 3 − 2 𝑦 2 + 4 𝑦 2 − 10𝑦
= 16 𝑦 3 − 4 𝑦 2 + 4 𝑦 2 − 10𝑦
𝟑
𝟏
Solution: 𝟏𝟔 𝒚𝟑 − 𝟒 𝒚𝟐 − 𝟏𝟎𝒚
49) (3x + 4)(3x–4)
Firsts
(3x)(3x)
Outers
(3x)(-4)
Inners
(4)(3x)
lasts
(4)(-4)
= 9x2 – 12x + 12x – 16
Solution: 9x2 - 16
1
1
51) (𝑥 + 4) (𝑥 − 4)
Firsts
(x)(x)
Outers
Inners
−1
1
(𝑥) ( ) ( ) (𝑥)
4
4
1
1
1
= 𝑥 2 − 4 𝑥 + 4 𝑥 − 16
Solution: 𝒙𝟐 −
𝟏
𝟏𝟔
lasts
1
4
−1
4
( )( )
#53 – 70: Multiply, write the answer in descending order
53) (2x–3)(x2 + 4x – 2)
= (2x)(x2) + (2x)(4x) + (2x)(-2) + (-3)(x2) + (-3)(4x) + (-3)(-2)
= 2x3 + 8x2 – 4x – 3x2 – 12x + 6
= 2x3 + 8x2 – 3x2 – 4x – 12x + 6
Solution: 2x3 + 5x2 – 16x + 6
55) (x – 4)(3x2 +x – 2)
= x(3x2) + x(x) + x(-2) + (-4)(3x2) + (-4)(x) + (-4)(-2)
= 3x3 + x2 – 2x – 12x2 – 4x + 8
Solution: 3x3 – 11x2 – 6x + 8
57) (5y + 1)(3y2 + 4y –3)
= (5y)(3y2) + (5y)(4y) + (5y)(-3) + (1)(3y2) + (1)(4y) + (1)(-3)
= 15y3 + 20y2 – 15y + 3y2 + 4y – 3
Solution: 15y3 + 23y2 – 11y - 3
59) (4z –3)(3z2+5z+4)
`
= (4z)(3z2) + (4z)(5z) + (4z)(4) + (-3)(3z2) + (-3)(5z) + (-3)(4)
= 12z3 + 20z2 + 16z – 9z2 – 15z – 12
Solution: 12z3 + 11z2 + z - 12
61) (4x2 +5x+1)(2x–7)
= (4x2)(2x) + (4x2)(-7) + 5x(2x) + (5x)(-7) + (1)(2x) + (1)(-7)
= 8x3 – 28x2 + 10x2 – 35x + 2x – 7
Solution: 8x3 – 18x2 – 33x - 7
63) (6x2 –7x–1)(3x–4)
= (6x2)(3x) + (6x2)(-4) + (-7x)(3x) + (-7x)(-4) + (-1)(3x) + (-1)(-4)
= 18x3 – 24x2 – 21x2 +28x – 3x + 4
Solution: 18x3 – 45x2 + 25x + 4
65) (x2 + 6x +2)(x2 + 5x +2)
= (x2)(x2) + (x2)(5x) + (x2)(2) + (6x)(x2) + (6x)(5x) + (6x)(2) + (2)(x2) + (2)(5x)+2(2)
= x4 + 5x3 + 2x2 + 6x3 + 30x2 + 12x + 2x2 + 10x + 4
= x4 + 5x3 + 6x3 + 2x2 + 30x2 + 2x2 + 12x + 10x + 4
Solution: x4 + 11x3 + 34x2 + 22x + 4
67) (5x2 + 3x + 1)(3x2 + 7x –1)
= (5x2)(3x2) + (5x2)(7x) + (5x2)(-1) + (3x)(3x2) + (3x)(7x) + (3x)(-1) + 1(3x2) + 1(7x) + 1(-1)
= 15x4 + 35x3 – 5x2 + 9x3 + 21x2 – 3x + 3x2 + 7x – 1
Solution: 15x4 + 44x3 + 19x2 + 4x – 1
69) (-7x2 + 2x –4)(3x2 –5x –3)
= (-7x2)(3x2) + (-7x2)(-5x) + (-7x2)(-3) + (2x)(3x2) + (2x)(-5x) + (2x)(-3) + (-4)(3x2) + (-4)(-5x) + (-4)(-3)
= -21x4 + 35x3 + 21x2 + 6x3 – 10x2 – 6x – 12x2 + 20x + 12
= -21x4 + 35x3 + 6x3 + 21x2 – 10x2 – 12x2 – 6x + 20x + 12
Solution: -21x4 + 41x3 – x2 + 14x + 12
#71 – 88: Multiply the “differences of squares”, try to be efficient in your work
71) (x + 3)(x–3)
I only need to do firsts and lasts as the outers and inners will cancel.
= (x)(x) + (3)(-3)
Solution x2 - 9
73) (3x + 5)(3x–5)
I only need to do firsts and lasts as the outers and inners will cancel.
= (3x)(3x) + (5)(-5)
Solution: 9x2 - 25
75) (4x3 + 7)(4x3 -7)
I only need to do firsts and lasts as the outers and inners will cancel.
= (4x3)(4x3) + (7)(-7)
Solution: 16x6 – 49
2
2
77) (𝑥 + 3) (𝑥 − 3)
I only need to do firsts and lasts as the outers and inners will cancel.
2
3
−2
3
= (x)(x) + ( ) ( )
𝟒
Solution: 𝒙𝟐 − 𝟗
2
2
79) (3𝑦 + 7) (3𝑦 − 7)
I only need to do firsts and lasts as the outers and inners will cancel.
2
−2
= (3𝑦)(3𝑦) + (7) ( 7 )
𝟒
Solution: 𝟗𝒚𝟐 − 𝟒𝟗
81) (5 + 2y3)(5-2y3)
I only need to do firsts and lasts as the outers and inners will cancel.
= (5)(5) + (2y3)(-2y3)
Solution: 25 – 4y6
83) (x2 – 5)(x2+5)
I only need to do firsts and lasts as the outers and inners will cancel.
= (x2)(x2) + (-5)(5)
Solution: x4 - 25
85) (3x – 4)(3x+4)
I only need to do firsts and lasts as the outers and inners will cancel.
= (3x)(3x) + (-4)(4)
Solution: 9x2 - 16
87) (5 – 2x)(5+2x)
I only need to do firsts and lasts as the outers and inners will cancel.
= (5)(5) + (-2x)(2x)
Solution: 25 – 4x2
#89 – 106: Multiply the squares, write the answer in descending form
89) (x + 3)(x+3)
Firsts
(x)(x)
Outers
(x)(3)
Inners
(3)(x)
lasts
(3)(3)
Inners
(5)(3x)
lasts
(5)(5)
= x2 + 3x + 3x + 9
Solution: x2 + 6x + 9
91) (3x + 5)(3x+5)
Firsts
(3x)(3x)
Outers
(3x)(5)
= 9x2 + 15x + 15x + 25
Solution: 9x2 + 30x + 25
93) (4x3 – 7)(4x3 –7)
Firsts
Outers
3
3
(4x )(4x ) (4x3)(-7)
Inners
(-7)(4x3 )
= 16x6 – 28x3 – 28x3 + 49
Solution: 16x6 – 56x3 + 49
lasts
(-7)(-7)
2
2
95) (𝑥 − 3) (𝑥 − 3)
−2
−2
−2
−2
= (𝑥)(𝑥) + 𝑥 ( 3 ) + ( 3 ) 𝑥 + ( 3 ) ( 3 )
2
2
4
= 𝑥2 − 3 𝑥 − 3 𝑥 + 9
𝟒
𝟑
𝟒
𝟗
Solution: 𝒙𝟐 − 𝒙 +
2
2
97) (3𝑦 + 7) (3𝑦 + 7)
Firsts
Outers
Inners
lasts
2
2
2
2
(3𝑦)(3𝑦)
(3𝑦) ( ) ( ) (3𝑦) (7) (7)
7
7
6
6
4
= 9𝑦 2 + 7 𝑦 + 7 𝑦 + 49
Solution: 𝟗𝒚𝟐 +
𝟏𝟐
𝒚
𝟕
+
𝟒
𝟒𝟗
99) (5 – 2y3)(5–2y3)
= (5)(5) + (5)(-2y3) + (-2y3)(5) + (-2y3)(-2y3)
= 25 – 10y3 – 10y3 + 4y6
Solution: 4y6 – 20y3 + 25
101) (x2 + 5)(x2+5)
= (x2)(x2) + (x2)(5) + 5(x2) + (5)(5)
= x4 + 5x2 + 5x2 + 25
Solution: x4 + 10x2 + 25
103) (3x – 4)(3x–4)
= (3x)(3x) + (3x)(-4) + (-4)(3x) + (-4)(-4)
= 9x2 – 12x – 12x + 16
Solution: 9x2 – 24x + 16
105) (5 +2x)(5+2x)
= 25 + 10x+ 10x + 4x2
Solution: 4x2 + 20x + 25
#107 – 120: Simplify, write the answer in descending form
107) (3x + 5)2
= (3x+5)(3x+5)
= 9x2 + 15x + 15x + 25
Solution: 9x2 + 30x + 25
109) (4x +1)2
= (4x+1)(4x+1)
= 16x2 + 4x + 4x + 1
Solution: 16x2 + 8x + 1
111) (3x – 5)2
= (3x – 5)(3x – 5)
= 9x2 – 15x – 15x + 25
Solution: 9x2 – 30x + 25
113) (4x –1)2
= (4x – 1)(4x – 1)
= 16x2 – 4x – 4x + 1
Solution: 16x2 – 8x + 1
115) (3 + 7y)2
=(3 + 7y)(3+7y)
= 9 + 21y + 21y + 49y2
Solution: 49y2 + 42y + 9
117) (2 + x)2
= (2 + x)(2 + x)
= 4 + 2x + 2x + x2
Solution: x2 + 4x + 4
119) (3 – 7y)2
= (3 – 7y)(3 – 7y)
= 9 – 21y – 21y + 49y2
Solution: 49y2 – 42y + 9
121) (2 – x)2
= (2 – x)(2 – x)
= 4 – 2x – 2x + x2
Solution: x2 – 4x + 4
#123-138: Multiply and simplify
123) (2x + 3y)(4x – 2y)
= (2x)(4x) + (2x)(-2y) + (3y)(4x) + (3y)(-2y)
= 8x2 – 4xy + 12xy – 6y2
Solution: 8x2 +8xy – 6y2
125) 3xy(2x + 3y)
= (3xy)(2x) + (3xy)(3y)
Solution: 6x2y + 9xy2
127) 8x2(4xy – 3y)
= (8x2)(4xy) + (8x2)(-3y)
Solution: 32x3y - 24x2y
129) (x2y3 + 2)(x2y3 -2)
= (x2y3)(x2y3) + (x2y3)(-2) + 2(x2y3) + (2)(-2)
= x4y6 – 2x2y3 + 2x2y3 – 4
Solution: x4y6 – 4
131) (3x – 4y)2
= (3x- 4y)(3x – 4y)
= (3x)(3x) + (3x)(-4y) + (-4y)(3x) + (-4y)(-4y)
= 9x2 – 12xy – 12xy + 16y2
Solution: 9x2 – 24xy + 16y2
133) (7 – 2xy)2
= (7 – 2xy)(7 – 2xy)
= 49 – 14xy - 14xy + 4x2y2
Solution: 4x2y2 – 28xy + 49
135) (5x – 2y)(3xy)
= (5x)(3xy) + (-2y)(3xy)
Solution 15x2y - 6xy2
137) (5ab + 1)(3 + 2ab)
= (5ab)(3) + (5ab)(2ab) + (1)(3) + 1(2ab)
= 15ab + 10a2b2 + 3 + 2ab
Solution: 10a2b2 + 17ab + 3