172 
… 4.5‹Products of polynomials.
„‹Please find the following products.
ƒS

# (2b)(-3b) = -6b’,ˆ‹3y(6-4y) = 3y(6) + 3y(-4y) = 18y - 12y
‚S
To multiply a monomial times a monomial, use the commutative and
associative properties of multiplication to group the numerical factors
and the variable factors in seperate paren‡s.

#ˆˆ‰(2b)(-3b)‹=‹(2)(-3)(bωb)

Then use the rules for multiplying signed numbers to multiply the
ˆˆˆˆˆŠm‹nŠm+n
numerical factors, and use the rule, a ωa = aŠ, to multiply the

#variables.ˆˆ‹= (-6)b’ = -6b’

#To multiply 3y(6 - 4y ), first use the distributive property to obtain,

#ˆˆˆ‰(3y)(6) + (3y)(-4y)

then treat each term of this expression as a monomial times a monomial.

#ˆˆˆˆŠ18y - 12yŽ
 1
#ˆˆ‰ Find the product‹(14aŽ)(-2a).


#A)‹12aˆ‰B)‹-28a“ˆ‹C)‹26a£ˆ D)‹† of ‡

#ˆŠ(14aŽ)(-2a) = (14)(-2)(aŽωa)‹=‹-28aŽ’‹=‹-28a“
€ B
 2
#ˆˆ‰ Find the product‹(-5x’)(-3x).


#A)‹15xŒŒˆŠ B)‹-15xŽˆ‹C)‹-15£ˆ D)‹† of ‡

#ˆŠ(-5x’)(-3x) = (-5)(-3)(x’ωx)‹=Š15x’’‹=Š15xŒŒ
€ A
 3
ˆˆ‰ Find the product‹2z(4z + 2)


#A)‹16z + 4ˆ‹B)‹8z + 4ˆ C)‹8z+ 4z‰ D)‹† of ‡


#ˆˆ2z(4z + 2)‹=‹(2z)(4z) + (2z)(2)‹=‹8z + 4z
€ C
 4
ˆˆ‰ Find the product -5p(5 - 4p) .


#A) -25p + 4pˆB) -25p + 20pŠ C) -25p- 20pŠ D)‹† of thes


#ˆŠ -5p(5 - 4p)‹=‹(-5p)(5) + (-5p)(-4p)‹=‹-25p + 20p
€ B
„‹Multiply the following binomials.
ƒˆ(2x + 3)(x - 1)‹=‹(2x + 3)ωx + (2x + 3)(-1)

ˆˆŠ=‹(2xωx + 3x + (2x)(-1) + 3(-1)

#ˆˆ‹= 2x + 3x - 2x - 3‹=‹2x + x - 3
‚S
To multiply two binomials such as (2x + 3) and (x - 1), use the
distributive property to multiply (2x + 3) times each term in the second
paren‡s.
ˆˆˆ‰ (2x + 3)(x - 1)
ˆˆˆŠ = (2x + 3)ωx + (2x + 3)(-1)

Then use the distributive property again.

ˆˆ‰ = (2x)(x) + 3ωx + 2x(-1) + 3(-1)

Finally, simplify each term and combine like terms.

#ˆˆ‰ = 2x+ 3x - 2x - 3‹=‹2x+ x - 3
 5
ˆˆˆ‹Multiply‹(x + 3)(3x - 4)

#A) 3x+ 5x - 12Š B) 2x- 12x + 5Š C) x- 5x + 12Š D) † of ‡
ˆˆˆ (x + 3)(3x - 4)

‰= (x + 3)3x + (x + 3)(-4)Š =Š xω3x + 3ω3x + x(-4) + 3(-4)


#ˆˆ‹= 3x+ 9x - 4x - 12Š =Š 3x+ 5x - 12
€ A
 6
ˆˆˆ‹Multiply‹(4y - 2x)(5y + x)

#A) 10y- 6xy + xŠ B) x- 4xy + y‰C) 20y- 6xy - 2x‰D)‹of
ˆˆˆˆˆˆˆˆˆŠ ‡
ˆˆˆ (4y - 2x)(5y + x)

= 14y - 2x)(5y) + (4y - 2x)ωxŠ=‹4y(5y + (-2x)(5y) + (4y)(x) + (-2x)ωx

#ˆ‰ =‹20y- 10xy + 4xy - 2x‹=‹20y- 6xy - 2x
€ C
 7
#ˆˆˆ‰Multiply‹(m - 3)
ˆˆˆˆˆˆˆˆˆŠ †
#A) m- 6m + 9ˆ‹B) m- 9ˆ‰ C) m+ 9ˆ‰ D)‹of
ˆˆˆˆˆˆˆˆˆŠ ‡

#ˆˆˆˆŠ (m - 3)

ˆ‰= (m - 3)(m - 3)Š=Š (m - 3)m + (m - 3)(-3)

#ˆˆ‹= m- 3m - 3m + 9‰=‰m- 6m + 9
€ A
 8
#ˆˆˆ‰Multiply‹(p + 2)Ž

ˆˆˆˆˆˆˆˆˆŠ†
#A) pŽ+ 8ˆ‰B) pŽ+ 6p+ 12p + 8ˆC) pŽ- 8ˆ‹D)‹of
ˆˆˆˆˆˆˆˆˆŠ‡

#ˆˆˆˆŠ (p + 2)Ž

ˆˆˆŠ= (p + 2)(p + 2)(p + 2)

#ˆ = (p+ 4p + 4)(p + 2)Š=ŠpŽ+ 4p+ 4p + 2p+ 8p + 8

#ˆˆˆ‰= pŽ+ 6p+ 12p + 8
€ B
„‹Multiply the following expressions.
ƒ
#ˆˆˆŠ (x + 3)(x- 2x + 3)

#ˆ‰= xŽ- 2x+ 3x + 3x- 6x + 9Š =Š xŽ+ x- 3x + 9
‚S
#To multiply an expression such asŠ (x + 3)(x- 2x + 3) just multiply
each term in the first group of paren‡s times each term in the second
group of paren‡s.

#ˆˆ‹(x)(x) + x(-2x) + xω3 + 3ωx+ 3(-2x) + 3ω3

#Then simplify individual terms.Š xŽ- 2x+ 3x + 3x- 6x + 9

#Finally, collect like terms.ˆxŽ+ x- 3x + 9
 9
#ˆˆˆMultiply‹(2x - 3)(x+ 3x - 5)

ˆˆˆˆˆˆˆˆˆ‰†
#A) 3x+6xˆB) 8xŽ- 3x+ x - 5‰ C) 2xŽ+ 3x- 19x + 15‰D) of
ˆˆˆˆˆˆˆˆˆ‰‡

#ˆˆˆ‰ (2x - 3)(x+ 3x - 5)


#‰= (2x)(x) + (2x)(3x) + (2x)(-5) + (-3)x+ (-3)(3x) + (-3)(-5)

#ˆ= 2xŽ+ 6x- 10x - 3x- 9x + 15‰=‰2xŽ+ 3x- 19x + 15
€ C
 10
#ˆˆˆMultiply‹(a + 4)(a- 3a+ 1)

ˆˆˆˆˆˆˆˆˆŠ †
#A) a- 6ˆ B) aŽ- 3a+ 9ˆŠ C) a- 6a+ 12ˆŠ D) of
ˆˆˆˆˆˆˆˆˆŠ ‡

#ˆˆˆŠ(a + 4)(a- 3a+ 1)


#ˆˆ= aωa+ a(-3a) + aω1 + 4ωa+ 4(-3a) + 4ω1

#‰ = a- 3aŽ+ a + 4a- 12a+ 4Š=Ša+ 4a-3aŽ- 12a+ a + 4
€ D