MULTIPLYING A POLYNOMIAL BY A MONOMIAL

Using distributive property of multiplication, you can multiply a polynomial (a + b + c) by a monomial (a) as shown below.   

Example 1 : 

Multiply. 

2(3x2 + 5x + 7)

Solution : 

=  2(3x2 + 5x + 7)

Distribute 2.

=  2(3x2) + 2(5x) + 2(7)

Multiply. 

=  6x2 + 10x + 14

Example 2 : 

Multiply. 

-5(2x3 - 5x2 + 7x - 3)

Solution : 

=  -5(2x3 - 5x2 + 7x - 3)

Distribute -5.

=  -5(2x3) - 5(-5x2) - 5(7x) - 5(-3)

Multiply. 

=  -10x3 + 25x2 - 35x + 15 

Example 3 : 

Multiply. 

2x2y(4x - y)

Solution : 

=  2x2y(4x - 3y)

Distribute 2x2y.

=  2x2y(4x) + 2x2y(-3y)

Group like terms together. 

=  (⋅ 4)(x⋅ x)(y) + 2(-3)(x2)(y ⋅ y)

Use the Product of Powers Property. 

8x2 + 1y - 6x2y1 + 1

=  8x3y - 6x2y2

Example 4 : 

Multiply. 

5a(a2b + 3b2)

Solution : 

=  5a(a2b + 3b2)

Distribute 5a.

=  5a(a2b) + 5a(3b2)

Group like terms together. 

=  5(a ⋅ a2)b + (5 ⋅ 3)ab2

Use the Product of Powers Property. 

=  5a1 + 2b + 15ab2

=  5a3b + 15ab2

Example 5 : 

Multiply. 

3ab(5a2 + b)

Solution : 

=  3ab(5a2 + b)

Distribute 3ab.

=  3ab(5a2) + 3ab(b)

Group like terms together. 

=  (3 ⋅ 5)(a ⋅ a2)b + 3a(b ⋅ b)

Use the Product of Powers Property. 

=  15a1 + 2b + 3ab1 + 1

=  15a3b + 3ab2

Example 6 : 

Multiply. 

-2a2b3(3ab2 - a2b)

Solution : 

=  -2a2b3(3ab2 - a2b)

Distribute -2a2b3.

=  -2a2b3(3ab2) - 2a2b3(-a2b)

Group like terms together. 

=  -(2 ⋅ 3)(a⋅ a)(b⋅ b2) - (2 ⋅ -1)(a⋅ a2)(b⋅ b)

Use the Product of Powers Property. 

=  -6a2 + 1b3 + 2 - (-2)a2 + 2b3 + 1

=  -6a3b5 + 2a4b4

Example 7 : 

Multiply. 

2xy(x2y + xz - xy)

Solution : 

=  2xy(x2y + xz - xy)

Distribute 2xy.

=  2xy(x2y) + 2xy(xz) + 2xy(-xy)

Group like terms together. 

=  2(x ⋅ x2)(y ⋅ y) + 2(x ⋅ x)yz + 2(-1)(x ⋅ x)(y ⋅ y)

Use the Product of Powers Property. 

=  2x1 + 2y1 + 1 + 2x1 + 1yz - 2x1 + 1y1 + 1

=  2x3y2 + 2x2yz - 2x2y2

Example 8 : 

Multiply. 

2a2b(ab2 - a2b + ab)

Solution : 

=  2a2b(ab2 - a2b + ab)

Distribute 2a2b.

=  2a2b(ab2) + 2a2b(-a2b) + 2a2b(ab)

Group like terms together. 

=  2(a⋅ a)(b ⋅ b2) + (2 ⋅ -1)(a⋅ a2)(b ⋅ b) + 2(a⋅ a)(b ⋅ b)

Use the Product of Powers Property. 

=  2a2 + 1b1 + 2 - 2a2 + 2b1 + 1 + 2a2 + 1b1 + 1

=  2a3b3 - 2a4b2 + 2a3b2

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